Standard string-polytope lattice-point conjecture for classical groups

Let $G$ be a complex classical group, let $\lambda\in\Lambda^+$, and let $\underline{w_0}^\mathrm{std}$ be the standard reduced decomposition of the longest word of the Weyl group of $G$. Write $Q_{\underline{w_0}^\mathrm{std}}(\lambda)$ for the associated string polytope. A polytope is a lattice polytope when all of its vertices lie in the ambient lattice.

Standard string-polytope conjecture. The polytope $Q_{\underline{w_0}^\mathrm{std}}(\lambda)$ is a lattice polytope if and only if at least one of the following holds: $G=\mathrm{SL}_{n+1}$; $G=\mathrm{SO}_{2n+1}$ and $\langle\lambda,\alpha_n^\vee\rangle\in2\mathbb{Z}$; $G=\mathrm{Sp}_{2n}$; or $G=\mathrm{SO}_{2n}$ and either $\langle\lambda,\alpha_{n-1}^\vee\rangle+\langle\lambda,\alpha_n^\vee\rangle\in2\mathbb{Z}$ or $n<4$.

This conjecture gives a precise integrality criterion for standard string polytopes of complex classical groups, extending the known results in types $\mathsf{A}_n$, $\mathsf{B}_n$, and $\mathsf{C}_n$. The type $\mathsf{D}_n$ case and the only-if direction in type $\mathsf{B}_n$ remain difficult; the source reports counterexamples to the earlier general Alexeev–Brion conjecture for higher-rank special linear groups.