Mutation Sequences along Weaves and Amalgamation of Braid Varieties

Let $p,q$ be positive braids with Demazure products $u,v$. The endpoint stratum $\mathrm{Conf}(p,q)_{u,v}$ of the double Bott-Samelson cell splits as $\mathrm{Conf}(u,v) \times X(p) \times X(q^{\mathrm{op}})$, a double Bruhat cell times two braid varieties. We prove that the stratum's cluster struct