Leclerc's conjecture on the cluster algebra of Richardson varieties

Let $v\leq w$ be elements indexing an open Richardson variety $\mathring{\mathcal{R}}_{v,w}$, and let $\mathbf{w}$ be the chosen word defining Leclerc's seed $\Sigma_{v,\mathbf{w}}^{\operatorname{Lec}}$. Write $\mathcal{A}(\Sigma_{v,\mathbf{w}}^{\operatorname{Lec}})$ for the associated cluster algebra and $\mathbb{C}[\mathring{\mathcal{R}}_{v,w}]$ for the coordinate ring of the open Richardson variety. Leclerc's conjecture. The cluster algebra

$$\mathcal{A}(\Sigma_{v,\mathbf{w}}^{\operatorname{Lec}})$$

is equal to

$$\mathbb{C}[\mathring{\mathcal{R}}_{v,w}].$$

Leclerc proved this equality in certain special cases, while the general equality is the conjectural assertion studied here.