Conjecture on the joint spectrum of higher Gaudin Hamiltonians and \mathfrak{gl}_{M|N} spaces

Let $\boldsymbol\lambda=(\lambda^{(1)},\dots,\lambda^{(n)})$ be a typical sequence of polynomial $\mathfrak{gl}_{M|N}$ weights, let $\boldsymbol z=(z_1,\dots,z_n)$ be a sequence of distinct complex numbers, and let $\boldsymbol T=(T_1,\dots,T_{M+N})$ be the corresponding polynomials. Set $L(\boldsymbol\lambda)=\bigotimes_{k=1}^n L(\lambda^{(k)})$, and let $\mathcal B(\boldsymbol\lambda)$ be the commutative algebra of higher Gaudin Hamiltonians. For a parity sequence $\boldsymbol s$, write $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}_{\lambda^{(\boldsymbol s,\infty)}}$ for the corresponding singular weight subspace. Joint-spectrum conjecture. The algebra $\mathcal B(\boldsymbol\lambda)$ has a simple joint spectrum in $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$. Eigenvectors of $\mathcal B(\boldsymbol\lambda)$ in $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}_{\lambda^{(\boldsymbol s,\infty)}}$, up to multiplication by a nonzero constant, are in bijective correspondence with $\mathfrak{gl}_{M|N}$ spaces of rational functions $W$ satisfying $T_i^W=T_i$ and $\lambda^{(\boldsymbol s,\infty)}_W=\lambda^{(\boldsymbol s,\infty)}$. Under this correspondence, for every $k=1,\dots,n$, the eigenvalue of the Gaudin Hamiltonian $\mathcal H_k$ is given by the stated Bethe-ansatz formula, with $\boldsymbol t$ represented by any $k$-admissible $\boldsymbol y$ in $\beta(\mathcal F(W))$. The conjecture extends the known polynomial $\mathfrak{gl}_M$ correspondence to the superalgebra setting; the paper does not establish it.