Elias's acyclicity and unique extrema conjecture for affine reduced-word graphs

For an affine permutation $w\in \widetilde{\mathfrak{S}}_n$, let $\mathcal{G}(w)$ be the directed graph whose vertices are commutation classes of reduced words for $w$, with edges oriented by the braid relations $s_is_{i+1}s_i\to s_{i+1}s_is_{i+1}$, where indices are taken modulo $n$. Elias's conjecture. For any $w\in \widetilde{\mathfrak{S}}_n$, the directed graph $\mathcal{G}(w)$ is acyclic with a unique source vertex and a unique sink vertex. This conjecture concerns the global structure of directed braid-move graphs for affine permutations; the supplied text does not state whether it has been resolved.