Generalised Delta square conjecture, valley version

Let $m,n,k$ be nonnegative integers, let $h_m$ and $e_j$ be the complete homogeneous and elementary symmetric functions, and let $\Delta_f$ be the Delta operator. Let $\omega$ be the standard involution, $p_n$ the power-sum symmetric function, and $\mathsf{LSQ}(m,n)^{\bullet k}$ the set of labelled valley-decorated square paths of size $m+n$ with $m$ zero labels and $k$ decorations. For $\pi$ in this set, write $\mathsf{dinv}(\pi)$, $\mathsf{area}(\pi)$, and $x^\pi$ for its associated statistics and monomial weight.

Generalised Delta square conjecture, valley version.

$$\frac{[n-k]_q}{[n]_q} \Delta_{h_m} \Delta_{e_{n-k}} \omega(p_n) = \sum_{\pi \in \mathsf{LSQ}(m,n)^{\bullet k}} q^{\mathsf{dinv}(\pi)} t^{\mathsf{area}(\pi)} x^\pi.$$

The paper reports computational evidence through $n=6$ and notes that this valley version differs from the rise version by replacing the $t$-analogue factor with a $q$-analogue factor. It remains open.