Leclerc–Ing quiver conjecture for Richardson varieties

Let $G=\operatorname{SL}_n(\mathbb{C})$, let $v\leq w$ be elements of the Weyl group $W$, and let $Q_{\operatorname{Lec}}$ and $Q_{\operatorname{Ing}}$ be the quivers constructed from Leclerc's and Ingermanson's seeds, respectively, with vertex set $J_{\mathbf{v}}^\circ$ when the irreducible-factor conjecture holds. Leclerc–Ing quiver conjecture. The quivers $Q_{\operatorname{Lec}}$ and $Q_{\operatorname{Ing}}$ are isomorphic; assuming the irreducible-factor conjecture, this isomorphism is given by naturally identifying both vertex sets with $J_{\mathbf{v}}^\circ$; and, for each $v\leq w\in W$, these quivers are locally acyclic. The conjecture is supported by extensive computational evidence, but no resolution is supplied here.