Let $\pi$ denote a saturated set of weights. Let $S\_q(\pi)$ denote the associated generalised $q$-Schur algebra. I was wondering if the following claim is true:


Claim: The algebra $S\_q(\pi)$ is Koszul if and only if the decomposition numbers of $S\_q(\pi)$ are given by the associated Kazhdan-Lusztig polynomials.


In characteristic zero and `sufficiently large primes' I believe this is true by work of Andersen-Jantzen-Soergel and Riche. But is it true or believed to be true over fields of arbitrary characteristic? 



Is Koszulity equivalent to the Lusztig character formula holding?
Since Geordie already started the shameless self-promotion: I believe you can apply the results of section 1 of [Canonical bases and higher representation theory](https://arxiv.org/abs/1209.0051) to show that this will happen whenever the graded version of this $q$-Schur algebra (which you may need to believe [a conjecture of Riche and Williamson](https://arxiv.org/abs/1512.08296) to define) is Morita equivalent to a positively graded algebra. So asking for Koszulity is, as Geordie points out, actually overkill.


That said, I think Koszulity will likely follow in this case: an argument like in [Notes on parameters of quiver Hecke algebras](http://arxiv.org/pdf/1204.1776.pdf) by Kashiwara should show that this positivity will only happen if the base change from characteristic 0 to characteristic $p$ sends simples to simples, in which case the numerical criterion shows that Koszulity is preserved.


