Edge graph of the polytope of a Bruhat interval

Question

Let $\Gamma$ be a Coxeter group on some generating set $S$, with reflection representation $V$. Then $\Gamma$ has two standard partial orders, the weak and strong Bruhat orders.

Moreover, if $\lambda \in V$ is chosen generically (any free orbit will do), then the covering relations in weak order are given exactly by the edge graph of the convex hull of $\Gamma\cdot \lambda$.

Let $[u,v]$ be an interval in strong Bruhat order. Has the edge graph of the polytope $hull([u,v]\cdot \lambda)$ been studied?

For example, the polytope $hull([123,321] \cdot (1,2,3))$ is a hexagon, and the strong Bruhat cover $231 > 132$ defines an edge through the middle of this hexagon, so not a weak cover. Whereas the polytope $hull([132,321] \cdot (1,2,3))$ is a trapezoid, one edge of which connects $231$ and $132$.

EDIT: perhaps I should admit the geometry here. If $\Gamma$ is a Weyl group of a Lie group $G$ -- and I am happy to make this assumption, albeit I want $G$ Kac-Moody -- and $V$ the corresponding weight lattice, and $\lambda$ a dominant weight, then $hull(W\cdot \lambda)$ is the moment polytope for $G/B$ bearing the Borel-Weil line bundle ${\mathcal L}_\lambda$. Within $G/B$ we have the Richardson variety $\overline{BuB}/B \cap \overline{B_- vB}/B$, and $hull([u,v]\cdot \lambda)$ is the moment polytope of that.

Answer 1

For type $A$, see Appendix A of Kodama-Williams. They prove that all the edges $Hull([u,v]) \cdot \lambda$ correspond to strong Burhat covers. They also show that $Hull([u,v]) \cdot \lambda$ is the Minkowski sum of the polytopes $c_i Hull([u,v]) \cdot \alpha_i$, where $\lambda = \sum c_i \alpha_i$ is the decomposition of $\lambda$ into simple roots.

Skimming the proofs, I think the second sentence should still be true in other types. The first sentence, however, uses the fact that weak and strong order on $S_n/(S_k \times S_{n-k})$ coincide, so I think this should not generalize.