Joint symmetry conjecture for q,t-rectangle numbers

For $a,b\geq 0$, define the q,t-rectangle numbers

$$S_{a,b}(q,t)=\sum_{w\in\mathcal{R}^{\mathrm{word}}(\mathrm{N}^a\mathrm{E}^b)}q^{\mathsf{area}^*_{b,-a}(w)}t^{\mathsf{area}^*_{b,-a}(\operatorname{sw}_{b,-a}^-(w))}.$$

Here $\mathsf{area}^*_{b,-a}$ is the adjusted area statistic defined using the minimum of the $(b,-a)$-levels. Joint symmetry conjecture. For all $a,b$,

$$S_{a,b}(q,t)=S_{a,b}(t,q).$$

The conjecture is known when $a=b$, where a stronger identity was proved in the cited literature; the general rectangular case remains conjectural in the source.