The brick-polytope quotient conjecture for the longest element

Let $Q$ be a sorting and alternating word in a Coxeter group with longest element $\omega_\circ$, let $\mathsf{Brick}(Q,\omega_\circ)$ be the associated brick polytope, and let $\equiv_{Q,\omega}$ be the subword complex equivalence. Brick-polytope quotient conjecture. If $Q$ is sorting and alternating, then the oriented graph of $\mathsf{Brick}(Q,\omega_\circ)$ is isomorphic to the Hasse diagram of a lattice quotient of the weak order. This is the specialization of the brick-polyhedron realization conjecture to brick polytopes and extends the cited results on brick algebras; the source gives no resolution status.