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?