The new square conjecture for decorated Dyck paths

For $n\geq 1$, let $\mathcal{LSQ}_n^E$ be the set of decorated labelled square paths of size $n$. For $(P,r)\in\mathcal{LSQ}_n^E$, let $\mathsf{area}(P,r)$ and $\mathsf{dinv}(P,r)$ be the area and diagonal-inversion statistics defined above, let $\sigma(P,r)$ be the permutation obtained by reading labels along diagonals, and let $\mathsf{ides}(\sigma(P,r))=\mathsf{Des}(\sigma(P,r)^{-1})$. For $S\subseteq\{1,2,\dots,n-1\}$, let $Q_{S,n}$ be the Gessel fundamental quasisymmetric function of degree $n$ indexed by $S$.

New square conjecture. For every $n\geq 1$,

$$\Delta_{e_{n-1}}e_n=\sum_{(P,r)\in\mathcal{LSQ}_n^E}q^{\mathsf{dinv}(P,r)}t^{\mathsf{area}(P,r)}Q_{\mathsf{ides}(\sigma(P,r)),n}.$$

This conjecture gives a combinatorial expression for a delta-operator specialization in terms of decorated labelled square paths. The supplied passage gives no evidence that it has been proved or disproved.