Partial Schutzenberger action conjecture for the affine cactus group

Let $\overline{M_{0,n+2}^{comp}}$ be the compact real moduli space, and let the eigenlines of Bethe subalgebras on a tensor product of Kirillov–Reshetikhin representations form the associated spectrum, equipped with its Kirillov–Reshetikhin crystal structure. The generators of the fundamental groupoid are paths between neighboring components of the open stratum crossing codimension-one strata; they correspond to longest elements of connected Dynkin subdiagrams in the affine Dynkin diagram $\tilde{A}_{n+1}$. Partial Schutzenberger action conjecture. These generators act on the Kirillov–Reshetikhin crystal by partial Schutzenberger involutions assigned to connected subdiagrams of $\tilde{A}_{n+1}$. This would identify the monodromy of the compact Bethe-subalgebra spectrum with standard crystal-theoretic involutions; the source gives no resolution.