Let $G$ be a semisimple algebraic group over $\mathbb{C}$ and for a highest weight $\lambda$, denote by $V_{\lambda}^w$ the Demazure module associated with $\lambda$ and $w$. More precisely, $V_{\lambda}^w=U(\mathfrak{b}).v_{w \lambda}$. There are many nice descriptions of these modules (sections of line bundles restricted to Schubert varieties, and more concretely the Demazure character formula).

If $P(V_{\lambda}^w)$ denotes the set of weights which appear in the Demazure module, I can consider the convex hull in $X^*(T) \otimes_{\mathbb{Z}} \mathbb{R}$. The extremal vertices of the resulting polytope will be $w'(\lambda)$ for $w' \leq w$ via an argument of Bernstein, Gelfand and Gelfand.

My questions generally run as follows: Is there a nice description of the facets of the resulting polytope? Is it true that if $F$ is a facet of this polytope that there exists a coweight $\check{\Lambda_i}$ such that for all $v \in F$ we have $\langle v, \check{\Lambda_i} \rangle =n$ for $n \in \mathbb{Z}$? Are they pseudo-Weyl polytopes in the sense of Kamnitzer (MV Cycles and Polytopes)?

I would be grateful for any knowledge in this direction.