Prove symmetry of the bigraded rational parking-function polynomial

Prove that the bigraded rational parking-function polynomial $PF_{m,n}(q,t)$ is symmetric under exchanging the variables $q$ and $t$ for the relevant coprime parameters $m$ and $n$.

Background

For the full regular singularity, the paper identifies the proposed Hitchin-system polynomial with the rational parking-function polynomial $PF_{m,n}(q,t)$. The polynomial records the area and dinv statistics of labeled rational Dyck paths and is central to the paper’s combinatorial construction.

The authors explicitly state that the resulting two-variable polynomial is conjectured to have $q,t$ symmetry, but no proof is supplied.