Non-coprime rational parking-function Shuffle Conjecture

Let $(m,n)$ be a coprime pair of positive integers and let $k\in\mathbb{N}$. Let $\operatorname{Park}_{km,kn}$ be the set of parking functions in the $km\times kn$ lattice, and let $\mathbf{e}_{km,kn}$ be the operator obtained from Algorithm~ by taking $f=e_k$. Let $\operatorname{area}(\pi)$, $\operatorname{dinv}(\pi)$, and $\operatorname{ides}(\pi)$ be the associated parking-function statistics.

Non-coprime rational Shuffle Conjecture. For all coprime pairs of positive integers $(m,n)$ and any $k\in\mathbb{N}$,

$$\mathbf{e}_{km,kn}\cdot(-\mathbf{1})^{k(n+1)}=\sum_{\pi\in\operatorname{Park}_{km,kn}}t^{\operatorname{area}(\pi)}q^{\operatorname{dinv}(\pi)}F_{\operatorname{ides}(\pi)}.$$

This is the coarsest proposed extension from coprime rational parking functions to arbitrary pairs with prescribed gcd. The text presents it as part of a sequence of conjectures and gives no resolution evidence.