Irreducibility and genus conjecture for quantum KdV spectral curves

For each $k\geq 1$, let $\Sigma_{k,m}$, for $m\geq 1$, be the weight-$k$ spectral curve

$$\Sigma_{k,m}:=\left\{(\sigma,\rho)\in\mathbb{C}^2:\ \det_{\widetilde{\Lambda}_k}(K_m(\sigma)-\rho)=0\right\}.$$

Here $\mathscr{P}_k$ denotes the set of partitions of weight $k$, and $\ell(\lambda)$ denotes the length of a partition $\lambda$. Irreducibility and genus conjecture. For any fixed $k\geq 1$, the curves $\Sigma_{k,m}$, $m\geq 1$, are irreducible, their geometric genus $g(\Sigma_{k,m})$ is independent of $m$, and

$$g(\Sigma_{k,m})=(k-1)\left|\mathscr{P}_k\right|+1-\sum_{\lambda\in\mathscr{P}_k}\ell(\lambda).$$

This conjecture predicts a uniform geometric description of the spectral curves associated with the quantum KdV hierarchy, extending the computed genus values for $\Sigma_{k,1}$ and relating the genus to the combinatorics of partitions of $k$.