Alexeev–Brion conjecture on integrality of string polytopes

Let $G=SL_{n+1}$, let $\underline{w_0}$ be a reduced decomposition of the longest word of the Weyl group, and let $\Lambda^+$ denote the set of dominant weights.

Alexeev–Brion conjecture. For every reduced decomposition $\underline{w_0}$ and every $\lambda\in\Lambda^+$, the string polytope $\mathcal{Q}_{\underline{w_0}}(\lambda)$ is a lattice polytope.

The paper states that this conjecture is false, so its asserted universal integrality fails for non-standard reduced decompositions. Alexeev and Brion are the attributed authors of the formulation; the candidate is therefore refuted.