Fraser–Sherman-Bennett conjecture on quasi-equivalence of relabeled plabic cluster structures

The source concerns open positroid varieties and the cluster structures determined by relabeled plabic graphs. For a relabeled graph $G^\rho$ and an ordinary plabic graph $H$ with the same trip permutation, write $\Sigma^T_{G^\rho}$ and $\Sigma^T_H$ for their target-labeled seeds; two such seeds are related by a quasi-cluster transformation when they are connected by mutations together with the permitted Laurent-monomial transformations in frozen variables. Quasi-equivalence conjecture. If $G^\rho$ is a relabeled plabic graph satisfying the conditions that make it determine a cluster structure on $\widetilde \Pi^{\circ}_\pi$, and $H$ is a plabic graph with trip permutation $\pi$, then $\Sigma^T_{G^\rho}$ and $\Sigma^T_H$ are related by a quasi-cluster transformation. This predicts that the different target cluster structures on the same open positroid variety have the same underlying structure up to the natural frozen-variable rescalings; the paper verifies important special cases, while the general assertion remains open.