The large-$l$ counting conjecture for Gaudin Bethe solutions

Let $d_0=d_1$ and let $\operatorname{Part}(d_0)$ denote the number of partitions of $d_0$. Large-$l$ Gaudin counting conjecture. For generic $l,m$, the Gaudin Bethe Ansatz system has, up to permutations of coordinates, exactly $\operatorname{Part}(d_0)$ solutions, all isolated and non-degenerate; for fixed $d_0$ there is $C\in\mathbb R$ such that $|l|,|m|>C$ implies genericity, and the solutions have the stated partition-indexed asymptotics as $l\to\infty$. The claim is part of the paper's large-parameter analysis and is not resolved in the supplied text.