Etingof's conjecture on pure cactus actions and crystals

Let $\CB_{\ul{\l}}$ be the set of highest elements in the tensor product $\CB_{\l_1}\otimes\cdots\otimes\CB_{\l_n}$ of $\fg$-crystals, and let $B_{\ul{\l}}(\ul{z})$ be the set of Bethe vectors. The pure cactus group $PJ_n$ acts on both sets: on Bethe vectors by monodromy and on crystals by the crystal commutor. Etingof's conjecture. The two $PJ_n$-actions are isomorphic. The conjecture compares Bethe-vector monodromy with the coboundary-category action on crystals.