The cluster variety conjecture for open Richardson varieties

Let $G$ be a simple complex algebraic group with Lie algebra $\mathfrak{g}$, let $B$ and $B_-$ be opposite Borel subgroups, and let $u\leq v$ be elements of the Weyl group. For the Schubert cells $\mathring X_v:=BvB/B$ and opposite Schubert cells $\mathring X^u:=B_-uB/B$ in the flag variety $G/B$, define the open Richardson variety

$$R^u_v=\mathring X_v\cap\mathring X^u.$$

Refined open Richardson conjecture. The open Richardson variety $R^u_v$ is a cluster variety satisfying the Louise property and of full rank.

The refinement would place open Richardson varieties within the class of cluster varieties for which the paper proves strong cohomological and point-count consequences. The supplied text gives no resolution of this conjecture; it is proposed after the weaker folklore conjecture and remains open.