Fraser's quasi-cluster transformation conjecture for Richardson twists

Let $v\leq w$ and let $\tau_{v,w}:\mathring{\mathcal{R}}_{v,w}\to\mathring{\mathcal{R}}_{v,w}$ be the right twist map. A quasi-cluster transformation means a sequence of cluster mutations followed by rescaling by Laurent monomials in frozen variables. Let $\mathcal{A}(\Sigma_{v,\mathbf{w}}^{\operatorname{Ing}})$ and $\mathcal{A}(\Sigma_{v,\mathbf{w}}^{\operatorname{Lec}})$ denote the cluster algebras associated with Ingermanson's and Leclerc's seeds, respectively. Richardson twist conjecture. The twist map $\tau_{v,w}$ is a quasi-cluster transformation. Consequently, any cluster of $\mathcal{A}(\Sigma_{v,\mathbf{w}}^{\operatorname{Ing}})$ is related to any cluster of $\mathcal{A}(\Sigma_{v,\mathbf{w}}^{\operatorname{Lec}})$ by a sequence of mutations and rescaling by Laurent monomials in frozen variables. The source notes that this extends the analogous conjecture for positroid varieties; its truth in the Richardson setting is not established in the supplied text.