Kleshchev–Martin conjecture on loop-free Ext-quivers of symmetric-group algebras

Let $r$ be a nonnegative integer, let $\Bbbk$ be the underlying field, and let $S_\Bbbk(r,r)$ be the classical Schur algebra. An $i$-faithful quasi-hereditary cover means a quasi-hereditary cover satisfying the corresponding $i$-faithfulness condition for some integer $i\geqslant 0$. Kleshchev–Martin conjecture. The ${\rm Ext}$-quiver of the group algebra of the symmetric group $\Bbbk S_r$ is loop-free if and only if the classical Schur algebra $S_\Bbbk(r,r)$ is an $i$-faithful quasi-hereditary cover for some $i\geqslant 0$. This conjecture concerns the relationship between self-extensions of simple modules and faithfulness of quasi-hereditary covers. It was previously known in some cases, including RoCK blocks, but the general conjecture remains open.