Joint symmetry conjecture for generalized q,t-Catalan polynomials

For $r,s\in\mathbb{Z}$ and $a,b\geq 0$, define

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

Joint symmetry conjecture. For all $r,s\in\mathbb{Z}$ and all $a,b\geq 0$,

$$C_{r,s,a,b}(q,t)=C_{r,s,a,b}(t,q).$$

The conjecture extends the usual joint symmetry of q,t-Catalan-type polynomials to arbitrary integer slopes and rectangular endpoints. The text notes that bijectivity of the relevant sweep map would imply only the weaker specialization at one variable equal to $1$; the general relation to nabla-type operators is presented as an open problem.