Completeness conjecture for the Gaudin model associated with \mathfrak{gl}$1|1$

Let $\mathfrak{gl}(1|1)$-weights $\bm\lambda^{(s)}$, $1\leqslant s\leqslant k$, be polynomial, let $L_{\bm\Lambda}(\bm b)$ be the corresponding tensor product representation, and let $(L_{\bm\Lambda}(\bm b))^{\mathrm{sing}}$ denote its singular subspace. Let $\varphi_{\bm\Lambda,\bm b}$ be the polynomial associated with these weights and evaluation parameters, and let $\mathscr H(x)$ be the Gaudin transfer matrix.

Completeness conjecture. The operator $\mathscr H(x)$ has simple spectrum on $(L_{\bm\Lambda}(\bm b))^{\mathrm{sing}}$. There is a bijection between the monic divisors $y$ of $\varphi_{\bm\Lambda,\bm b}$ and the eigenvectors $v$ of the Gaudin transfer matrices, up to multiplication by a non-zero constant, such that

$$\mathscr H(x)v=\mathcal E_{y,\bm\Lambda,\bm b}(x)v,$$

where $\mathcal E_{y,\bm\Lambda,\bm b}(x)$ is the eigenvalue defined by the Bethe-ansatz formula in the source.

This is the completeness statement for the Bethe ansatz: every relevant eigenvector is parametrized by a monic divisor of $\varphi_{\bm\Lambda,\bm b}$, and distinct divisors yield the simple spectrum. The parser provides no resolution evidence, so the conjecture is recorded as open.