The subclosedness conjecture for categories associated to preprojective algebra 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. Write $\mathscr{C}(\Pi/I_w)$ for the associated full subcategory of $\operatorname{mod} kQ$. Subclosedness conjecture. For every $w\in W$, the subcategory

$$\mathscr{C}(\Pi/I_w)$$

of $\operatorname{mod} kQ$ is subclosed. This is proved in the Dynkin case and is also verified when $Q$ has two vertices; the general case remains open.