But, behind the scenes, the PM-in-waiting will be hearing loud and clear that reform can't be put off forever - including from his own inner circle.
Two of his economic advisers have previously spoken against the triple lock, warning that it is unsustainable.
We've spoken to experts about the options available to the man who will likely be prime minister in a matter of weeks...
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Why are experts worried it's unsustainable?
Under the triple lock, the state pension rises each year by whichever is highest out of wage growth, inflation or 2.5%.
When the triple lock was introduced in 2012, estimates suggested it would cost around £5.2bn a year by 2029-30 and see the state pension rise by an average of 0.2 percentage points above earnings growth.
But inflation has been significantly more volatile than expected - and this has triggered some significant annual uplifts.
While inflation has eased in the last couple of years, earnings growth has shot up - resulting in another boost for pensioners, but another dent to the public purse.
The government's financial watchdog, the Office for Budget Responsibility, has forecast the government to spend £15.5bn more a year on the state pension than if it were increased in line with earnings alone by 2030 - around three times higher than initial expectations.
In the 2025-26 tax year, state pension payments totalled around £146.1bn - for context, we spent around £62.2bn on defence in 2024-25.
The Department for Work and Pensions expects state pension spending to hit £169bn by 2030.
And, considering the population is living longer and giving birth to fewer children to grow into pension-funding taxpayers, rising government spending on the state pension is likely to continue well beyond 2030.
Burnham advisers don't like triple lock
In July 2025, then OBR chair Richard Hughes said "uncertainty around the operation of the triple lock" is an "important source of fiscal risk".
He added at the time: "When you project trends in both pension spending and health and other age-related spending forward, the UK public finances are in an unsustainable position in the long run.
"The UK cannot afford the array of promises that are displayed to the public if you leave those unchanged based on a reasonable assumption about growth rates in the economy and in tax revenues."
And what is Hughes doing now? He is one of the two economic advisers to Andy Burnham we mentioned at the start of this piece.
The other is Lord O'Neill - the economist, co-president of the Northern Powerhouse Partnership and a former Goldman Sachs Asset Management chairman.
Last year, he was widely reported as calling the triple lock "bonkers" and suggested the state pension should be means tested.
Thinktanks from across the political spectrum have also warned it is not sustainable.
One of those is the Resolution Foundation, a thinktank once led by the current pensions minister Torsten Bell, that falls on the centre-left of the political spectrum.
The Foundation has said: "It is impossible to justify why our generosity to the older population should be a function of economic volatility.
"As well as being unfair, it is also not fiscally sustainable for the state pension to rise forever by more than the earnings of a typical worker."
What do voters think?
Despite the criticism from experts, scrapping the policy, or even changing it, is not popular with the voting public - which might explain why no prime minister wants to touch it.
A YouGov poll from April found that 37% of people opposed making any changes to the triple lock policy, while only 26% supported changes.
Unsurprisingly, older people were more likely to oppose any reforms to the policy, while younger people were more likely to support changes.
A separate study of 2,000 people by financial advice firm AJ Bell found just 6% of Britons wanted the triple lock to be scrapped, while 38% thought it should be made permanent.
Again, it found a significant generational divide with more than two-thirds of baby boomers (aged 60+) wanting to see the triple lock stay, versus just 14% of Gen Z (18-29-year-olds) and 22% of millennials (30 to 45-year-olds).
Could it be scrapped? And does Burnham have other options?
Tom Selby, director of public policy at AJ Bell, thinks any path towards the triple lock being "retired" eventually requires a clear explanation of the trade-offs to voters.
"Assuming the Treasury does not want spending as a share of GDP on state pensions to continue ballooning - squeezing the ability to spend elsewhere or reduce the tax burden on the working population - there are two main levers available to control costs: the amount people receive from the state in retirement and the age at which they receive it," Selby said.
Considering the government's fiscal position, changes will have to be made to the state pension at some point. So if scrapping the triple lock isn't a plausible solution, then what other changes could Burnham make to reduce costs?
Researchers at the National Institute of Economic and Social Research (NIESR) have proposed replacing the triple lock with a "living standards lock". Under their proposal, the state pension would continue to be uprated in line with inflation each year to protect its real value, with additional earnings-linked increases only in years when real earnings are growing and have reached a new high.
Another lever that could be pulled is the state pension age, which is undergoing statutory review.
The state pension age is 66 for both men and women, but the age is rising: for those born on or after 6 April 1960, it is gradually increasing towards 68.
Pensions expert Charlene Young from AJ Bell told Money: "Raising the state pension age without meaningful improvements in healthy life expectancy risks pushing more people into working-age benefits later in life.
"In that scenario, the government could simply end up shifting costs from one part of the welfare system to another, rather than delivering substantial savings overall."
Should we copy German and Swedish models?
Germany has already proposed raising its retirement age to 70 by the early 2090s, and scrapping the option of retiring early at 63 after making 45 years of contribution payments.
The proposal from the country's pension commission is one of many sweeping reforms being suggested to future-proof the German pension system for an ageing population.
It also suggested setting up a fund modelled on the Swedish pension system, with mandatory contributions by workers and employers that would be invested in financial assets as a complement to the current pay-as-you-go system.
The idea is that this would help to fund the government's rising pension costs.
Sweden made it mandatory for 2.5% of Swedes' pensionable income to be invested into funds of their choice or defaulted into a government-run fund 100% invested in equities until you turn 55.
When the "premium pension" was first introduced in 2000, it was criticised for being too complicated, but the default fund used has delivered an average return of about 7.4% a year.
Nouran Moustafa, owner of financial advice firm Roxton Wealth, told the Money blog: "A UK long-term state investment fund deserves serious consideration.
"Run independently and transparently, it could invest national wealth in global markets and create another source of pension funding. It is not magic, but neither is pretending tax receipts will stretch forever."
AJ Bell's Young added that a German-style sovereign wealth fund would represent a "more radical approach" - and certainly isn't a "silver bullet".
"While investing in capital markets to help fund future pension costs has its attractions, building assets at a scale capable of making a material difference would take decades and could still require higher contributions from today's workers and taxpayers, who are already funding current pensions," she said.
"Germany's own plans suggest the projected contribution from the fund would remain relatively small compared with the overall cost of state pensions, indicating this is unlikely to be a silver bullet."
Is Australian means testing the answer?
Another option could be to means test the state pension. This system is already in place in Australia, where the amount retirees receive is based on their income and the value of their assets.
For some of the wealthiest, this means they do not get any state pension payments.
The system has been criticised for quickly becoming difficult to navigate, with various benefits and schemes potentially affecting entitlement or whether a person qualifies at all.
Jamie Jenkins, director of policy at Royal London, told us: "Some countries, such as Australia, means test their state pension, but they also have much more generous provision through workplace pension saving, so it isn't easily comparable."
Antonia Medlicott, managing director at personal finance and investment platform Investing Insiders, said this could be something that Burnham considers, but warned it could be unpopular.
"He has publicly committed to keeping the triple lock, but he may consider means testing the state pension, limiting the amount wealthier people receive, or excluding them from it altogether; after all, he has spoken about taxing the wealthiest more, whether that's through pay or assets.
"However, this would cause uproar in many areas of society."
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A taxation fix?
Other options could include charging income tax on state pensions.
The full state pension is already likely to rise above the personal allowance next year, with it currently sitting at £12,547.60 a year - just £22.40 under the tax threshold.
We could see more pensioners dragged into this tax bracket over the next five years as the government has already announced that thresholds will be frozen until April 2031.
At the moment, pensioners will not have to pay small amounts of tax due from 2027-28, if the new or basic state pension exceeds that allowance.
But could Burnham change that?
Medlicott said: "Rumours suggest the state pension will be automatically taxed when it surpasses the personal allowance, which will cause anger, but could be the sweet spot he needs to find a solution that repairs public finances without taking away too many benefits from pensioners.
"Burnham may also consider protecting the triple lock and still controlling the cost to the treasury by freezing allowances for pensioners."
Jenkins added: "There is already some recognition that this is an odd way to proceed, paying the state pension on one hand, and taking tax on the other."
Targeting tax relief
There is also the option of targeting pension tax relief - a government top-up that refunds the income tax you would pay on earnings that are put into a pension.
The Pension Commission, an independent body reviewing the state of retirement in the UK, recently called pension tax relief "regressive".
Moustafa said this would be the worst option.
"That punishes people doing exactly what the government says it wants: saving for retirement. Reform should reward work, saving and long-term thinking, not make private provision less attractive," she said.
Young added: "Reducing pension tax relief could raise revenue in the short term, but it may also discourage retirement saving at a time when millions of people are already projected to fall short of the income they will need in retirement.
"Ultimately, that risks increasing future reliance on means tested state support, shifting costs back to the public purse through a different route."
Focus on private pensions
Another option could be to place greater emphasis on private pension provision to help reduce reliance on the state pension.
Young pointed out that recent proposed changes to salary sacrifice arrangements from 2029 are expected to reduce National Insurance savings for employers, potentially weakening incentives to support workplace pension saving, and something could be done to change that.
"Pensions are designed to support long-term saving, but repeated speculation around tax-free cash, pension tax relief and wider pension tax reforms can undermine trust in the system," she said.
"As we saw ahead of recent budgets, uncertainty prompted many savers to make rushed, and in some cases irreversible, decisions to access pension benefits for fear of losing valuable tax advantages. If policymakers want people to commit money to pensions for decades, they need confidence that the rules will remain broadly stable."
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One certain long-term problem
Away from all this uncertainty is one grim certainty - we as a nation are not saving enough for our retirement.
The Pension Commission has been quietly assessing the state of retirement saving in the UK and the key challenges facing the current system.
While it will publish its complete findings next spring, its interim report showed that 15 million people are currently undersaving for their retirement.
"The Pensions Commission sets out clearly the scale of the challenge: not enough people are saving for retirement, and many of those that are aren't saving enough," said pensions minister Torsten Bell.
"The commission warns that without action, millions more people could be at risk of becoming reliant on state support in retirement."
While we don't have its recommendations yet, the commission has said it expects working longer and over-50s returning to some form of work to be part of the solution.
"Working longer - and in particular reducing labour market inactivity among people in their fifties - is a necessary part of achieving adequate incomes in retirement," the commission has said.
But with the UK having a lower life expectancy than some countries with a younger retirement age, like Japan, we can't see the prospect of working longer being a popular solution either.
We could find out what Burnham will do - and whether he actually chooses to tackle the issue at all - in the next few weeks.
