The pmaj polyomino conjecture

Let $m,n\in\mathbb{N}$, let $\mathsf{LPP}(m+1,n+1)$ be the set of labelled parallelogram polyominoes of the indicated dimensions, and let $\mathsf{area}(P)$, $\mathsf{pmaj}(P)$ and $x^P$ denote their statistics and label monomial. The pmaj polyomino conjecture.

$$(qt)^{m+n+1}\Delta_{h_m}e_{n+1}=\sum_{P\in\mathsf{LPP}(m+1,n+1)}q^{\mathsf{area}(P)}t^{\mathsf{pmaj}(P)}x^P.$$

The source gives one proved scalar-product specialization as evidence, but the full identity is presented as a conjecture and remains open.