The pmaj formula for labelled parallelogram polyominoes

Let $m,n\geq 1$. Write $\mathsf{LP}(m,n)$ for the set of labelled parallelogram polyominoes of size $m\times n$, let $\mathsf{area}(P)$ and $\mathsf{pmaj}(P)$ denote their area and partially labelled major index statistics, and let $x^P$ be the product of the variables corresponding to the labels of $P$. Here $\Delta_{h_{m-1}}$ denotes the Delta operator associated with the complete homogeneous symmetric function $h_{m-1}$, and $e_n$ is the elementary symmetric function.

The pmaj formula. For $m\geq 1$ and $n\geq 1$,

$$\Delta_{h_{m-1}}e_n=\sum_{P\in\mathsf{LP}(m,n)}q^{\mathsf{area}(P)}t^{\mathsf{pmaj}(P)}x^P.$$

This is a proposed symmetric-function interpretation of labelled parallelogram polyominoes, suggested by computer verification; the source does not provide a proof or resolution.