Cluster compatibility conjecture for decorated reduced words

Let $G$ be the reductive group with Weyl group $W$, let $U$ be its maximal unipotent subgroup, and let $w_0$ be the longest element of $W$. For a decorated reduced word $\mathbb{I}$ of $w_0$, let ${\bf X}_{\mathbb{I}}$ be the associated set of $m=\dim(U)$ irreducible polynomials on $U$.

Cluster compatibility conjecture. For any decorated reduced word for $w_0$, the set ${\bf X}_{\mathbb{I}}$ is a cluster for $U$ mutation-equivalent to an initial cluster attached to any reduced word ${\bf i}$ for $w_0$.

This conjecture proposes that the toric coordinates arising from decorated reduced words fit into the existing cluster structure on the maximal unipotent subgroup. Its resolution is not given in the source.