BLZ-polynomial counting conjecture for highest-weight states

Let $\beta\in\mathbb C$ and $n\in\mathbb N$. Let $d_n^{(\beta)}$ denote the dimension or counting quantity defined in the source for the degree-$n$ states at momentum $\beta$, and let BLZ polynomials of level $n$ and momentum $\beta$ be the corresponding Wronskian polynomials. BLZ-polynomial counting conjecture. For every $(\beta,n)\in\mathbb C\times\mathbb N$, $d_n^{(\beta)}$ coincides with the number of distinct BLZ polynomials of level $n$ and momentum $\beta$. This is motivated by the ODE/IM interpretation in terms of states of the irreducible highest-weight representation; the source gives no resolution, so it remains open.