Touching Delta conjecture, valley version

Let $n,k,r$ be nonnegative integers, let $E_{n-k,r}$ be the symmetric function defined in the paper, and let $\Theta_{e_k}$ and $\nabla$ be the Theta and nabla operators. Let $\mathsf{LD}(n\backslash r)^{\bullet k}$ be the set of labelled valley-decorated Dyck paths with the prescribed parameter $r$; for $\pi$ in this set, write $\mathsf{dinv}(\pi)$, $\mathsf{area}(\pi)$, and $x^\pi$ for its associated statistics and monomial weight.

Touching Delta conjecture, valley version.

$$\Theta_{e_k} \nabla E_{n-k,r} = \sum_{\pi \in \mathsf{LD}(n \backslash r)^{\bullet k}} q^{\mathsf{dinv}(\pi)} t^{\mathsf{area}(\pi)} x^\pi.$$

This refinement is designed to imply the $m=0$ case of the valley Delta conjecture after summing over $r$. Its general validity remains open.