The level-rank duality Hilbert-series conjecture for SH$^c$ modules

Let $N$ and $M$ be the parameters of the level-rank duality, let $X$ be the partially ordered set defined by the relations in the source, and let $\mathcal{A}(X)$ be the set of all P-partitions over $X$. Write $x_i=\sum_{j=i}^{N}n_j$ and let $(a;q)_\infty=\prod_{k=0}^{\infty}(1-aq^k)$ denote the $q$-Pochhammer symbol. The level-rank duality Hilbert-series conjecture. The Hilbert series of the SH$^c$ module satisfies

$$\sum_{\lambda\in\mathcal{A}(X)}q^{|\lambda|}=\frac{(q^{N+M};q^{N+M})^{N-1}_{\infty}\prod_{i<j}(q^{x_i-x_j};q^{N+M})_{\infty}\prod_{i>j}(q^{N+M+x_i-x_j};q^{N+M})_{\infty}}{(q;q)_{\infty}^{N}}.$$

This conjecture extends the explicitly computed low-rank cases and predicts that the P-partition generating function agrees with the minimal-model character expression associated with level-rank duality. The general Jordan–Hölder set of the poset is not known, making the required P-partition count difficult; the status is therefore unresolved.