The toric quotient conjecture for hybrid fans

Let $W \subset \mathrm{GL}(V)$ be a finite reflection group preserving a lattice $M \subset V$ and an $M$-rational complete simplicial fan $\Sigma \subset V$. Let $X_\Sigma$ be the associated toric variety, and let $\Sigma_W$ denote the hybrid fan. Toric quotient conjecture. The quotient $X_\Sigma/W$ under the induced action of $W$ is isomorphic to the toric variety $X_{\Sigma_W}$ associated with the hybrid fan $\Sigma_W$. This would extend the previously known variety-level constructions for normal fans of weight polytopes and for simple non-degenerate $W$-symmetric polytopes to the potentially non-proper actions addressed by the hybrid fan construction.