But not in relative terms. What has happened is that the classes out of which people are not moving have shrunk. Increasingly it may not so much be education that counts as the unexaminable qualities associated with home background. But mention the reform of the law and you are in the realm of political controversy. There is some evidence from Scandinavia that socially mobile societies are fairer in their distribution of income.
Egalitarians therefore tend to be in favour of more ladders.
Private schooling is not of itself a key to increasing mobility since rates of social movement are similar across countries where fee-paying schools play widely differing roles. But middle-class parents think a good school matters. That is why they colonise catchment areas of schools doing well in league tables. Work for Demos by Ken Worpole showed how primary schools would have to be forced to take a mixed entry if the chance of a good start were to be made universally available - which could mean going back to the days of quotas for entry through ability bands. Imagine the furore.