Bender–Knuth monodromy conjecture for skew tableaux

Let $J_{n+1}$ be the cactus group associated with the split real moduli space, and let the caterpillar points index the eigenbases of Bethe subalgebras on a tensor product of skew representations of $Y(\mathfrak{gl}_n)$ with sufficiently distant real parameters. The paths along one-dimensional strata connecting caterpillar points give transformations of the set of semistandard skew tableaux. Bender–Knuth monodromy conjecture. These paths generate the fundamental group $J_{n+1}$, and the action of its generators on semistandard skew tableaux is given by Bender–Knuth involutions. This predicts a combinatorial description of the split-real monodromy action; the source gives no resolution.