Associahedron conjecture for mutation graphs of torus braids

Let $\beta$ be the $(n,2)$ torus braid, let $\Delta$ denote the positive half-twist, and consider the mutation graph whose vertices are equivalence classes of Demazure weaves $\beta\cdot\Delta\to\Delta$ and whose edges correspond to mutations. Torus-braid associahedron conjecture. This mutation graph is the $1$-skeleton of the $(n-1)$-dimensional associahedron. The conjecture identifies the weave mutation graph for these torus braids with the combinatorial exchange graph of the associahedron; the paper presents it as a conjectural extension of the pentagon example.