The \mathfrak{gl}_{1|1} divisor parametrization conjecture for Gaudin eigenvectors

Consider a tensor product $L(\boldsymbol\lambda)$ of typical polynomial $\mathfrak{gl}_{1|1}$ modules, with singular subspace $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$, and let $\mathcal N(T)$ be the polynomial defined from the corresponding data. Let $p,q$ denote the relevant weight parameters. \mathfrak{gl}_{1|1} divisor conjecture. The Gaudin Hamiltonians $\mathcal H_k$, $k=1,\dots,n$, have a simple joint spectrum in $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$. There is a one-to-one correspondence between monic divisors $y$ of $\mathcal N(T)$ of degree $l$ and joint eigenvectors $v$ of the Gaudin Hamiltonians of weight $(p-l,q+l)$, up to multiplication by a nonzero constant. Under this correspondence, $\mathcal H_kv=E_kv$ for $k=1,\dots,n$, where $E_k$ are given by the stated formula. This is the $\mathfrak{gl}_{1|1}$ specialization of the preceding conjecture, giving an explicit divisor parametrization; it remains conjectural in the paper.