I had some things on a hard drive that exploded, which would have supported my first pessimistic response to this, but I managed to retrieve some of them from an even older hard drive.
First there is a presentation from an "academic" called Francis Green. He was privately educated and now he makes a living from attacking independent schools. 35 minutes in (slide 16 in the pdf) he discusses ways to discourage parents from choosing independent schools through taxation and discriminatory university admissions.
https://www.lse.ac.uk/Events/Events-Assets/PDF/2019/01-LT/20190211-Engines-of-Privilege-presentation.pdf
He has written a book about it and probably numerous papers. In "The Necessity of Reforming Britain's Private Schools", which is attached below, he says:
"A reform strategy needs to be both feasible and effective. In Green and Kynaston (2019), we consider a number of proposals that fit this bill. On the one hand, one can attempt to diminish the demand for private schooling by parents, with the result that many schools would have to close or transfer over to the state sector. Taxing school fees is the most direct method, for example through the imposition of VAT; removing charitable status is another (though this also comes with considerable obstacles, and would not make much difference beyond the symbolic); imposing strong contextual admissions requirements on elite universities would also have a notable effect."
I don't know if this polemic was peer reviewed but the scholarship is "interesting". On page one for example he tells us that: "In academic terms, private schooling is shown to improve children’s performance in both low-stakes tests and high stakes public exams." In support of which he cites Smith-Woolley who in fact concludes that "...exam differences between school types are primarily due to the heritable characteristics involved in pupil admission." He also cites Crawford and Vignoles who do not support his statement; and Henderson who show "no private school advantage for attending an elite university". I stopped looking for anything that might support his assertion at this point.
https://discovery.ucl.ac.uk/id/eprint/10077819/7/Green_VoR_FORUM_61_2_web.pdf
So no, I don't think discrimination by school type is going to end. Cambridge is merely weighing up which is more problematic, "too many" privately educated students under its roof, or state-educated students lagging too far behind them.
Sorry about the length but it is a very interesting phenomenon.