The reduced Ext-quiver characterization conjecture for subcategories from preprojective quotients

Let $Q$ be a quiver, let $W$ be its Weyl group, let $\Pi$ be the associated preprojective algebra, and for $w\in W$ let $I_w$ be the corresponding ideal. For a full subcategory $\mathcal Z$ of $kQ$-modules, define its reduced $\operatorname{Ext}$-quiver to have the indecomposable objects of $\mathcal Z$ as vertices, with an arrow from $Y$ to $X$ when the simple $\mathcal Z$-module associated to $X$ is a direct summand of the socle of $\operatorname{Ext}^1_{kQ}(-,Y)$. Reduced Ext-quiver conjecture. The subcategory $\mathcal Z$ arises as $\mathscr{C}(\Pi/I_w)$ for some $w\in W$ if and only if it has finitely many indecomposables, is submodule-closed, and its reduced $\operatorname{Ext}$-quiver contains no cycles. This is proposed as the general description of the subcategories arising from the quotients; the paper proves the conjectural descriptions in the Dynkin case.