Perfect integrability for Gaudin algebras with arbitrary irregular parameter

Let $\mathfrak{g}$ be a Lie algebra, let $\mu\in\mathfrak{g}^*\cong\mathfrak{g}$, and choose $\mu\in\mathfrak b_- = \mathfrak n_-\oplus\mathfrak h$ after conjugation. Let $\mathfrak z_\mu(\mathfrak g)$ be the centralizer of $\mu$ in $\mathfrak g$, let $V_{\boldsymbol\lambda}$ be the relevant tensor-product representation, and define

$$\mathcal M_{\boldsymbol\lambda,\mu}:=\{v\in V_{\boldsymbol\lambda}\mid xv=0\text{ for all }x\in\mathfrak z_\mu(\mathfrak g)\cap\mathfrak n_+\}.$$

Let $\mathcal A_{\boldsymbol z,\mu}$ be the corresponding Gaudin algebra acting on this subspace. A finite-dimensional module is perfectly integrable if the algebra acts cyclically and its image in the endomorphism algebra is Frobenius. General-parameter perfect integrability conjecture. The $\mathcal A_{\boldsymbol z,\mu}$-module $\mathcal M_{\boldsymbol\lambda,\mu}$ is perfectly integrable. This extends the perfect-integrability expectation from regular parameters to arbitrary $\mu$ and uses the centralizer-defined subspace in place of the ordinary singular subspace; the source gives no resolution.