That study is really old @Newbutoldfather . It measures the results of students entering University in 2006.18 years ago.
OP gave a link to Cambridge's own published analysis from 2019. (Still 5 years out of date, but a full decade more recent than the study you link.)
https://www.cao.cam.ac.uk/admissions-research/app-research-papers-2020
You can download 'Analysis of student characteristics and attainment outcomes at the University of Cambridge' which was done in 2019.
At the time of the study, Cambridge students from private schools were 1.2 times more likely than students from state schools to get a first.
This is more statistically significant than the difference in the other direction 18 years ago which the study you link to describes.
Note that 18 years ago that difference was seen as evidence of discrimination against state students, and prompted Cambridge to explicitly change admission criteria to positively discriminate in favour of state candidates. (note: school type specifically, regardless of social class, wealth, or other privilege)
But now that this positive discrimination has caused the pendulum to swing the other way (and to a larger difference ) for some reason some people don't recognise this as evidence that the positive discrimination has gone too far.
@Marchesman gave us some more recent numbers from FOI requests, which show the difference increasing even more. She seems to have a specific interest and so has gathered information which she's kindly shared. Obviously FOI responses don't have to be published but I've taken them in good faith since anyone could make their own FOI request to Cambridge to verify them.
She told us that sudents from private schools were more likely than students from comprehensive schools to be awarded firsts by the following multiples from 2017 to 2023: 1.24 ,1.17, 1.19, 1.17, 1.23, 1.43, and 1.44.
Thankfully, Cambridge have now said that they will no longer specifically positively discriminate purely on school type (although of course other contextual factors will still be considered). Presumably due to Cambridge's own analysis of these statistics.
The other thread has some discussion about it.