The sorting-alternating quotient conjecture for subword complexes

Let $Q$ be a word and let $\omega$ be an element of a Coxeter group. For the non-empty subword complex $\mathcal{SC}(Q,\omega)$, let $[e,\omega]$ be the corresponding weak-order interval and let $\equiv_{Q,\omega}$ be the subword complex equivalence. The word $Q$ is sorting when it contains a reduced expression for the longest element $\omega_\circ$, and it is alternating when every pair of non-commuting generators alternates within $Q$. Sorting-alternating quotient conjecture. If $Q$ is sorting and alternating, then the Hasse diagram of the lattice quotient $[e,\omega]/\equiv_{Q,\omega}$ is isomorphic to the graph of extremal increasing flips between acyclic facets of $\mathcal{SC}(Q,\omega)$. In the sorting case all acyclic facets are strongly acyclic, so this specializes the alternating-word quotient conjecture; the source gives no resolution status.