The q-Schur Rock block conjecture

Let $q\in k^\times$, and let $p$ be the least natural number such that

$$1+q+\cdots+q^{p-1}=0.$$

Let $n,w$ be natural numbers with $n\geq w$. Let ${\cal Q}_p(n,w)$ be the quasi-hereditary subquotient of ${\cal D}_{A_\infty}(n,w)$ with poset $\Pi_p(n,w)$, and let $k{\bf B}_{\rho,w}^{{\cal S}_q}$ and $k{\bf B}_{\tau,w}^{{\cal S}_q}$ denote blocks of a $q$-Schur algebra of weight $w$. The q-Schur Rock block conjecture. The algebra ${\cal Q}_p(n,w)$ is Morita equivalent to any Rock block $k{\bf B}_{\rho,w}^{{\cal S}_q}$ of weight $w$; indeed, it is derived equivalent to any block $k{\bf B}_{\tau,w}^{{\cal S}_q}$ of weight $w$. This conjecture proposes a uniform quasi-hereditary model for weight-$w$ blocks, with Morita equivalence for Rock blocks and the stronger derived-equivalence assertion for arbitrary blocks; no resolution is given in the source.