Perfect integrability for regular quasi-periodic Gaudin spin chains

Let $\mathfrak{g}$ be a Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Cartan subgroup $H$, let $M$ be the finite-dimensional representation under consideration, and let $\mathscr B_{\mathfrak{g}}^\mu$ be the commutative Bethe algebra associated with $\mu$. A finite-dimensional module is perfectly integrable if the algebra acts cyclically and its image in the endomorphism algebra is Frobenius. Regular quasi-periodic integrability conjecture. If $\mu\in\mathfrak{h}^*$ (respectively, $\mu\in H$) is regular, then the $\mathscr B_{\mathfrak{g}}^\mu$-module $M$ is perfectly integrable. This extends the periodic case to regular quasi-periodic boundary conditions, where the Bethe algebra no longer commutes with the diagonal $\mathrm U(\mathfrak{g})$-action; the source gives no resolution.