There is loads of evidence to 'support that claim' - it's in the history books and it is happening today, sadly.
Okay, so you can't point me towards the loads of evidence
Not all is lost, I can point you towards this report, produced under the previous government in 2022
"The total rate has remained stable: “men and women who were born in the 1980s are no less likely to have moved to different class positions to those of their parents than men and women who were born at any time earlier in the 20th century...."
There are now fewer people at the bottom of the social ladder to make big leaps in social status, and more people starting life higher up, with less room to climb. Together, these changes mean that, although the total mobility rate has stayed the same, younger generations of men and women are more likely to experience downward mobility and less likely to experience upward mobility than their parents or grandparents. The surplus of upward over downward, though, is still positive. It had declined to 14 points in the youngest cohort, born in the 1970s"
There's also this discourse The Limits of ‘Opportunity’: Is There a Clear Labour or Conservative View of Social Mobility? - Maslen - 2024 - The Political Quarterly - Wiley Online Library on the different political views on mobility, note it discusses the conservative rigid, hierarchical gatekeeping on access to higher education via the 'elite pathway' into Oxbridge et al.
None of those references bear out what you've stated
Also, its inherently clear that true social mobility is contingent on the higher professional roles being available to move into in the first place - which is highly contingent on the long-term state of the economy, which has been in decline since 2008 and took a further hammering due to the Austerity loan we've all been paying from 2010.
Please add your thoughts on the above sources