Cluster algebra structure conjecture for braid varieties

Let $\eta$ be a braid word and let $X(\eta)$ denote the corresponding braid variety. A cluster algebra structure conjecture asserts that the coordinate ring of any braid variety $X(\eta)$ admits a structure of a cluster algebra. The exchange type of the mutable part of its defining quiver is preserved under Reidemeister II moves, Reidemeister III moves, and $\Delta$-conjugations of the braid word $\eta$. In addition, each such move gives rise to a quasi-cluster transformation. A positive stabilization adds one frozen vertex to the defining quiver, and a positive destabilization specializes one frozen variable to $1$. These claims are part of the paper's conjectural program relating braid varieties to cluster structures; the supplied passage does not state that they have been proved or refuted.