The diagonal-word generating-function corollary for parking functions

For a parking function $PF$, let $\operatorname{area}(PF)$, $\operatorname{dinv}(PF)$, and $\operatorname{ides}(PF)$ be its statistics, let $\mathcal{A}_c(\tau)$ be the set of parking functions with composition $c$ and diagonal word $\tau$, and define

$$\mathcal{C}_c^\tau=\sum_{PF\in\mathcal{A}_c(\tau)}t^{\operatorname{area}(PF)}q^{\operatorname{dinv}(PF)}Q_{\operatorname{ides}(PF)}.$$

The diagonal-word generating-function conjecture. For $b\leq a-1$ and every permutation $\tau$, one has

$$\mathcal{C}_{a,b}^\tau+\mathcal{C}_{b+1,a-1}^\tau=q\left(\mathcal{C}_{b,a}^\tau+\mathcal{C}_{a-1,b+1}^\tau\right).$$

This is presented as the generating-function corollary of the stronger bijection conjecture: the factor $q$ records the increase of $\operatorname{dinv}$, while preservation of the diagonal word permits the identity for each $\tau$. The source gives no resolution.