Toric Bruhat interval polytopes

For two elements $v$ and $w$ of the symmetric group $\mathfrak{S}_n$ with $v\leq w$ in Bruhat order, interval polytope $Q_{v,w}$ is convex hull points $(z(1),\ldots,z(n))\in \mathbb{R}^n$ z\leq w$. It known that moment map image Richardson variety $X^{v^{-1}}_{w^{-1}}$. We say \emph{toric} if corresponding $X_{w^{-1}}^{v^{-1}}$ a toric variety. show when toric, its combinatorial type determined by poset structure $[v,w]$ while this not true unless toric. are concerned problem (combinatorially equivalent to) cube because only smooth Boolean algebra. also give several sufficient conditions on for to be cube.