Berenstein–Kirillov quotient conjecture for the pure cactus group

Let $\fg=\mathfrak{gl}_N$, let the $V_{\l_i}$ be symmetric powers of the standard representation, and let $PJ_n$ act on the associated Gelfand–Tsetlin polytope through Bethe-vector monodromy. Let the Berenstein–Kirillov group act on that polytope by its piecewise-linear involutions. Berenstein–Kirillov quotient conjecture. The Berenstein–Kirillov group is a quotient of the pure cactus group $PJ_n$. The claim compares the cactus-group action arising from Gaudin theory with the previously defined Berenstein–Kirillov action.