Integrality conjecture for type A string polytopes

Let $G$ be of type $A_n$, let $\underline{w}_0$ be any reduced decomposition of the longest Weyl-group element, and let $\lambda$ be a weight. Write $Q_{\underline{w}_0}(\lambda)$ for the associated polytope, and let $\alpha_i^{\vee}$ denote the simple coroots. The polytope is integral when its vertices have integral coordinates. Integrality conjecture. For every reduced decomposition $\underline{w}_0$, the polytope $Q_{\underline{w}_0}(\lambda)$ is integral if and only if

$$\langle\lambda,\alpha_i^{\vee}\rangle\in\mathbb Z$$

for all $i$. The paper gives computational evidence in low-rank type A, including integrality for $G=\operatorname{SL}_n$ with $n\leq 5$, but does not state a resolution.