Anderson–Mirković conjecture for the MV-polytope crystal structure

Let $G$ be a complex connected reductive group, let $G^{\vee}$ be its Langlands dual group, and let $\mathcal{MV}$ denote the set of Mirković–Vilonen polytopes. For $j\in I$, let $AM_j:\mathcal{MV}\to\mathcal{MV}$ be the Anderson–Mirković operator defined from the pseudo-Weyl-polytope data, and let $\widetilde{f}_j$ be the Kashiwara operator.

Anderson–Mirković conjecture. For every MV polytope $P$ and every $j\in I$, $AM_j(P)$ is an MV polytope and

$$AM_j(P)=\widetilde{f}_j(P).$$

The conjecture proposes an explicit description of the crystal operator on MV polytopes, avoiding the recursive solution of tropical Plücker relations. It was disproved by a counterexample in type $C_2$; hence it is not an open conjecture in the stated generality.