Alternating subword complex equivalence conjecture

Let $Q$ be a word in the simple generators of a finite Coxeter group, let $\omega$ be a Coxeter-group element, and let $[e,\omega]$ be the corresponding interval in the weak order. Suppose that $Q$ is alternating, meaning that every pair of non-commuting generators $s,t$ alternates within $Q$. The subword complex equivalence relation $\equiv_{Q,\omega}$ on $[e,\omega]$ is defined by restricting the partition into sets of linear extensions to this interval. Alternating subword complex equivalence conjecture. If $Q$ is alternating, then $\equiv_{Q,\omega}$ is a lattice congruence of the interval $[e,\omega]$ of the weak order. This would identify the equivalence induced by alternating subword complexes with a lattice-theoretic quotient of the weak-order interval. The supplied text does not indicate whether the assertion has been proved or remains open.