The face-map conjecture for local-model toric strata

Let ${\rm Adm}^{\mathcal G}(\{\mu\})$ be the admissible set and let $|\Delta|$ be the divisor map from this poset to the face poset ${\mathscr F}(\sigma_{{\mathcal G},\mu})$. For $w\in{\rm Adm}(\{\mu\})$, define

$$\Lambda(w)=\{\bar\mu'\in\Lambda_{\{\mu\}}\mid w\leq t^{\bar\mu'}\}.$$

Let $|\Delta|^{\rm f}(w)$ be the smallest face of $\sigma_{{\mathcal G},\mu}$ containing the extremal rays corresponding to the elements of $\Lambda(w)$. The face-map conjecture. The map $|\Delta|$ coincides with the map $|\Delta|^{\rm f}$. This gives a purely combinatorial description of the divisor map; the source does not state a resolution or further cases in which it is known.