The weak BLZ conjecture for monster potentials

Let $N\geq 1$, and let $p(N)$ denote the number of integer partitions of $N$. Consider the BLZ system for the roots of an $N$-root monster potential, with parameters $\alpha$ and $L$.

Weak BLZ conjecture. For every $\alpha$ and $L$, the number of solutions of the BLZ system is at most $p(N)N!$. For generic values of $\alpha$ and $L$, the number of solutions is $p(N)N!$.

Modulo permutations of the roots, this predicts $p(N)$ monster potentials for generic parameters, matching the dimension of the level-$N$ subspace of the corresponding generic irreducible highest-weight Virasoro module. The claim is presented as an open consequence of the BLZ correspondence.