Perfect integrability of singular vectors for Gaudin Bethe algebras

Let $\mathfrak{g}$ be a Lie algebra, let $M$ be the finite-dimensional representation under consideration, let $M^{\mathrm{sing}}$ be its singular subspace, and let $\mathscr B_{\mathfrak{g}}$ be the corresponding commutative Bethe algebra. A finite-dimensional module is perfectly integrable if the algebra acts cyclically and its image in the endomorphism algebra is Frobenius. Perfect integrability conjecture. The $\mathscr B_{\mathfrak{g}}$-module $M^{\mathrm{sing}}$ is perfectly integrable. This is motivated by results for quantum spin chains and would relate the structure of Bethe algebras to Frobenius algebras and geometric Langlands constructions; the source presents it as an expected conjecture.