Acyclicity conjecture for the mutation graph of positive braids

A positive braid word is a word in the braid generators with only positive exponents, and its mutation graph has vertices given by equivalence classes of Demazure weaves and edges corresponding to mutations. Orient every mutation in the direction $(ss)s\to s(ss)$. Mutation-graph acyclicity conjecture. For any positive braid $\beta$, this orientation descends to the mutation graph of $\beta$, and the resulting oriented graph has no oriented cycles. This predicts a well-defined acyclic orientation on the mutation graph, supporting its interpretation as an exchange graph for a cluster algebra.