Gaudin monodromy conjectures for crystal tensor products

Let $\mathfrak{g}$ be a simple Lie algebra, let $V(\lambda_1),\ldots,V(\lambda_n)$ be irreducible representations, write $V({\underline{\lambda}})=V(\lambda_1)\otimes\cdots\otimes V(\lambda_n)$, and let $V({\underline{\lambda}})_\mu$ denote its $\mu$-weight space. Let $B(\lambda_i)$ be the normal crystal of highest weight $\lambda_i$, and let $vC_n=\pi_1^{S_n}(\overline F_n(\mathbb{R}))$, $\widetilde{C_n}=\pi_1^{S_n}(\overline M_{n+2}^{split}(\mathbb{R}))$, and $\widetilde{AC}_n=\pi_1^{S_n}(\overline M_{n+2}^{comp}(\mathbb{R}))$. The inhomogeneous and trigonometric Gaudin Hamiltonians have corresponding eigenlines in $V({\underline{\lambda}})_\mu$. Gaudin monodromy conjectures. (1) The monodromy action of $vC_n$ on the inhomogeneous Gaudin eigenlines in $V({\underline{\lambda}})_\mu$ matches the action of $vC_n$ on $B(\lambda_1)\otimes\cdots\otimes B(\lambda_n)$ given by crystal commutors and naive permutation of tensor factors. (2) The monodromy action of $\widetilde{C_n}$ on the trigonometric Gaudin eigenlines in $V({\underline{\lambda}})_\mu$ factors surjectively through the action of $vC_n$. (3) The monodromy action of $\widetilde{AC}_n$ on the trigonometric Gaudin eigenlines in $V({\underline{\lambda}})_\mu$ factors through the action of $vC_n$ and is given by crystal commutors and cyclic rotation of tensor factors. These conjectures relate monodromy of Gaudin eigenlines to the combinatorial actions on tensor products of crystals; the source says they are intended to be proved in future work, so their resolution is not established here.