The torsion-class criterion conjecture for preprojective ideals

Let $Q$ be a quiver with compatible ordering of its vertices, let $c=s_1s_2\cdots s_n$ be the corresponding Coxeter element, and let $W$ be its Weyl group. For $w\in W$, let $I_w$ be the associated ideal, let $\mathscr{C}(I_w)$ be the associated category, and let $\operatorname{sort}_c(w)$ denote the longest $c$-sortable prefix of $w$. Torsion-class criterion conjecture. The following conditions are equivalent: (1) the additive category generated by $\mathscr{C}(I_w)$ together with all non-preprojective indecomposable $kQ$-modules is a torsion class; (2) for every $i$ such that $\ell(ws_i)>\ell(w)$, $\operatorname{sort}_c(ws_i)$ is strictly longer than $\operatorname{sort}_c(w)$. The paper proves this in the Dynkin case and leaves the general equivalence conjectural.