Extended valley Delta square conjecture

Let $m,n,k$ be nonnegative integers. Write $\nabla$, $\Theta_{e_k}$, and $\Delta_{h_m}$ for the nabla, Theta, and Delta operators on symmetric functions, let $\omega$ be the standard involution on symmetric functions, and let $p_{n-k}$ be the power-sum symmetric function. For a labelled Schröder path in $\mathsf{LSQ}'(m,n)^{\bullet k}$, let $\mathsf{dinv}(\pi)$, $\mathsf{area}(\pi)$, and $x^{\pi}$ denote its diagonal inversion statistic, area statistic, and monomial weight, respectively. Extended valley Delta square conjecture.

$$\Delta_{h_m}\Theta_{e_k}\nabla \omega(p_{n-k})=\sum_{\pi\in \mathsf{LSQ}'(m,n)^{\bullet k}}q^{\mathsf{dinv}(\pi)}t^{\mathsf{area}(\pi)}x^{\pi}.$$

This is the Schröder, or square-path, analogue of the extended valley Delta conjecture, expressing a symmetric-function operator expression as a weighted generating function for labelled Schröder paths. Its resolution status is not specified in the supplied source context.