The t-statistic conjecture for gamma-parking functions

Let $\gamma$ and $\lambda$ be the parameters used to define the set $\operatorname{PF}^{\gamma}_{\lambda}$ of $\gamma$-parking functions, let $S$ be a subset of the area cells of $p$, and let $\eta(p)$ denote the indexing composition or partition associated with $p$. t-statistic conjecture. There exists a statistic $\mathsf{tstat}$ on pairs $(p,S)$, with $p\in\operatorname{PF}^{\gamma}_{\lambda}$ and $S$ a subset of the area cells of $p$, such that

$$\left.\Delta_{m_\gamma}\Xi e_{\lambda}\right\rvert_{q\rightarrow 1+u}=\sum_{p\in\operatorname{PF}^{\gamma}_{\lambda}}u^{\#S}t^{\mathsf{tstat}(p,S)}e_{\eta(p)}.$$

The statistic would refine the known $q=1+u$ positivity by supplying the missing $t$-weight and a combinatorial expansion through $\gamma$-parking functions. The text states that finding such a statistic remains open.