The compositional shuffle conjecture for $\nabla C_\alpha$

Let $\alpha\models n$ be a composition, let $C_\alpha[X;q]$ be the generalized Hall–Littlewood symmetric function indexed by $\alpha$, let $\mathcal{W}P_D$ denote the set of words associated with a Dyck path $D$, and let $\operatorname{touch}(D)$ record its diagonal touch-point composition. Write $\operatorname{area}(D)$ for the area of $D$ and $\operatorname{dinv}(w)$ for the diagonal inversion statistic of $w$. The compositional shuffle conjecture for $\nabla C_\alpha$. One has

$$\nabla\bigl(C_\alpha[X;q]\bigr)=\sum_{\operatorname{touch}(D)=\alpha}\sum_{w\in\mathcal{W}P_D}t^{\operatorname{area}(D)}q^{\operatorname{dinv}(w)}x^w.$$

This is the monomial expansion strengthening the touch-point coefficient identity for $C_\alpha$ and gives a proposed word-level refinement of the compositional shuffle conjecture. It is stated conjecturally in the source, and the general identity remains open there.