The partially labelled Dyck path Delta conjecture

Let $\operatorname{PLDP}(m,n)^{\star k}$ be the set of partially labelled Dyck paths of size $m+n+1$ with $n+1$ labels, $m$ blank labels that occur at valleys, and $k$ decorated rises. For $P$ in this set, let $\underline{\operatorname{area}}(P)$ and $\operatorname{pmaj}(P)$ be the associated statistics, and let $x^P$ denote the product of the variables $x_i$ over the nonzero labels of $P$.

Partially labelled Dyck path Delta conjecture. One has

$$\Delta_{h_m} \Delta'_{e_{n-k}} e_{n+1} = \sum_{P \in \operatorname{PLDP}(m,n)^{\star k}} q^{\underline{\operatorname{area}}(P)} t^{\operatorname{pmaj}(P)} x^P.$$

This is a partially labelled Dyck path formulation of the Delta conjecture. The paper establishes related cases and explains that the displayed identity is the conjectural combinatorial formula for the corresponding symmetric function.