Cluster-theoretic conjecture for splicing braid varieties

Let $\beta=\beta^1\beta^2$ be the braid and let $r_1$ and $w$ be as in the construction of the open subset $\mathcal{U}_{r_1,w}(\beta)\subseteq X(\beta)$. Assume that $\mathcal{U}_{r_1,w}(\beta)$ is nonempty. Let $Q_1$ and $Q_2$ be the ice quivers associated to the cluster structures on $\mathbb{C}[X(\underline{w^{-1}w_0}\beta^1)]$ and $\mathbb{C}[X(\beta^2\underline{w})]$, respectively, and write $Q^{\mathrm{uf}}$ for the quiver obtained from an ice quiver $Q$ by deleting its frozen vertices and all adjacent arrows. Splicing braid varieties conjecture. There exists a seed $\Sigma=(Q,\mathbf{x})$ for the cluster structure on $\mathbb{C}[X(\beta)]$ such that: (1) there are cluster variables $x_{a_1},\dots,x_{a_s}\in\mathbf{x}$ whose common nonvanishing locus is $\mathcal{U}_{r_1,w}(\beta)$; (2) $\mathcal{U}_{r_1,w}(\beta)$ admits a cluster structure with initial seed $\widehat{\Sigma}=(\widehat Q,\widehat{\mathbf{x}})$ obtained by freezing these variables; (3) $\widehat Q^{\mathrm{uf}}$ is mutation equivalent to the disjoint union of $\Q_1^{\mathrm{uf}}$ and $\Q_2^{\mathrm{uf}}$; and (4), with the natural product cluster structure on $\X(\underline{w^{-1}w_0}\beta^1)\times X(\beta^2\underline{w})$, the map

$$\Phi_{r_1,w}:\mathcal{U}_{r_1,w}(\beta)\to X(\underline{w^{-1}w_0}\beta^1)\times X(\beta^2\underline{w})$$

is a cluster quasi-isomorphism. This predicts that the cluster structure on the splicing open set is obtained by localizing the cluster structure on $X(\beta)$ and that its unfrozen quiver separates into the two braid-variety factors.