Conjecture on integrality of standard string polytopes for classical groups

Let $G$ be a complex classical group, let $\Lambda^+$ denote its dominant weights, and let $\underline{w_0}^{\mathrm{std}}$ be the standard reduced decomposition of the longest word of the Weyl group of $G$ stated in Littelmann's reference.

Integrality conjecture for standard string polytopes. The string polytope $\mathcal{Q}_{\underline{w_0}^{\mathrm{std}}}(\lambda)$ is a lattice polytope if and only if at least one of the following holds:

1. $G$ is of type $\mathsf{A}_n$;
2. $G$ is of type $\mathsf{B}_n$ and $\langle\lambda,\alpha_n^\vee\rangle\in 2\mathbb{Z}$;
3. $G$ is of type $\mathsf{C}_n$; or
4. $G$ is of type $\mathsf{D}_n$ and either $\langle\lambda,\alpha_{n-1}^\vee\rangle+\langle\lambda,\alpha_n^\vee\rangle\in 2\mathbb{Z}$ or $n<4$.

The conjecture is motivated by known results and calculations for classical types. Together with the paper's reflexivity result, it would yield a criterion for reflexivity of these standard string polytopes on partial flag varieties, but the integrality assertion itself is left open in the supplied text.