The Generalized Delta conjecture

Let $D$ be a partially labelled decorated Dyck path of size $n+m$ with $m$ zero labels, $n$ nonzero labels, and $k$ decorated rises. Its area word is $a(D)=a_1(D)\cdots a_{n+m}(D)$, where $a_i(D)$ counts the whole squares in the $i$-th row between the path and the main diagonal. The area and dinv statistics are

$$\mathsf{area}(D)=\sum_{i\notin\mathsf{DRise}(D)}a_i(D),$$

where decorated rises are excluded from the area, and $\mathsf{dinv}(D)$ counts primary and secondary inversions of the labelled area word. Write $\mathsf{PLD}(m,n)^{\ast k}$ for the set of such paths. Generalized Delta conjecture.

$$\Delta_{h_m}\Delta'_{e_{n-k-1}}e_n=\sum_{D\in\mathsf{PLD}(m,n)^{\ast k}}q^{\mathsf{dinv}(D)}t^{\mathsf{area}(D)}x^D.$$

This is a generalized form of the Delta conjecture, relating symmetric-function operators to weighted decorated Dyck paths. The source attributes it to Haglund, Remmel, and Wilson; its resolution status is not specified here.