Vertexwise admissibility conjecture for minuscule cocharacters

Let $\{\mu\}\subset X_*(T)$ be a $W$-conjugacy class of minuscule cocharacters, let $\bm{f}$ be a subfacet of the base alcove $\bm{a}$, and let $K$ be the associated parahoric subgroup. Write $\operatorname{Adm}_K(\{\mu\})$ for the parahoric admissible set and $\operatorname{Adm}^{\mathrm{vert}}_K(\{\mu\})$ for the intersection of the admissible conditions at all vertices of $\bm{f}$.

Vertexwise admissibility conjecture. The inclusion

$$\operatorname{Adm}_K(\{\mu\})\subseteq\operatorname{Adm}^{\mathrm{vert}}_K(\{\mu\})$$

is an equality.

The vertexwise set is automatically a superset of the parahoric admissible set. The conjecture asserts that checking admissibility at every vertex introduces no extra elements; the supplied source gives no resolution.