Generic solution-count conjecture for the quantum KdV oper system

Let $P_n(N)$ be the number of $n$-coloured partitions of $N$, let $n=\operatorname{rank}\mathfrak{g}$, and let $\operatorname{systemonc}$ denote the displayed system of equations for the oper data at level $N$. Let $(r,\hat{k})\in\mathfrak{h}\times(0,1)$ be the parameters of the model. Solution-count conjecture. The number of solutions of $\operatorname{systemonc}$ is at most $N!P_n(N)$. The set of parameters $(r,\hat{k})$ for which the number of solutions is exactly $N!P_n(N)$ is a generic subset of $\mathfrak{h}\times(0,1)$. This conjecture follows from the expected bijection between solutions of the system and level-$N$ quantum KdV states, together with the generic state count $P_n(N)$; its resolution is not supplied here.