The symmetry conjecture for the rational combinatorial Hilbert series

Let $m$ and $n$ be positive integers, let $\widetilde{S}_n^m$ be the set of $m$-stable affine permutations, and define the statistics

$$\operatorname{area}(\omega):=\frac{(m-1)(n-1)}{2}-\sum_i\operatorname{\mathcal{A}}_\omega(i),\qquad \operatorname{dinv}(\omega):=\frac{(m-1)(n-1)}{2}-\sum_i\operatorname{\mathcal{PS}}_\omega(i).$$

Define the combinatorial Hilbert series by

$$H_{m/n}(q,t):=\sum_{\omega\in\widetilde{S}_n^m}q^{\operatorname{area}(\omega)}t^{\operatorname{dinv}(\omega)}.$$

The combinatorial Hilbert-series symmetry conjecture. The series is symmetric in $q$ and $t$ for all $m$ and $n$:

$$H_{m/n}(q,t)=H_{m/n}(t,q).$$

The claim is presented as a conjecture and is related to the rational-slope generalization of the $q,t$-Catalan symmetry problem; the source does not give a resolution for all $m,n$.