Telegraph has been siren on this recently
The tax in not particularly efficient, compared to say a proper land value tax, but even in its present form its nowadays a huge earner for the Treasury, so change is less than likely.
But would the key argument, that cutting it would release lots of homes for younger people from older people downsizing, hold water?
No. The flaw in the argument is the belief that houses prices would remain the same. Never ague from a price is the economics saying, you also have to look at the demand effects.
A price cut from a tax cut would encourage more sales, but also more purchasers, we have a huge shortage of housing, especially right size housing for older people. The additional purchasers would push prices back up again. In technical terms the housing market is supply inelastic. This would mean the impact on released supply would be small, and also incidentally the reduction in the Stamp Duty tax take. This line of thinking suggests that some reform of stamp duty might have some minor effect at the margin, and be revenue neutral, if for example you lowered it for age categories likely to downsize, and raised it for categories likely to upsize. However a tax break based on age would be politically toxic. It isnt going to happen.