The Delta Conjecture for generalized coinvariant algebras

Let $\mathcal{LD}_n$ be the set of labeled Dyck paths of size $n$. For $P\in\mathcal{LD}_n$, let $a_i(P)$, $\ell_i(P)$, $\operatorname{area}(P)$, $\operatorname{dinv}(P)$, $d_i(P)$, $\operatorname{Val}(P)$, and $\mathbf{x}^P$ denote the associated area sequence, labels, statistics, contractible valleys, and monomial, respectively. Let $e_n(\mathbf{x})$ be the elementary symmetric function and $\Delta'_{e_{k-1}}$ the primed Delta operator; write $\{z^{n-k}\}$ for extraction of the coefficient of $z^{n-k}$. For positive integers $k\leq n$, The Delta Conjecture.

$$\Delta'_{e_{k-1}}e_n(\mathbf{x})=\{z^{n-k}\}\left[\sum_{P\in\mathcal{LD}_n}q^{\operatorname{dinv}(P)}t^{\operatorname{area}(P)}\prod_{i:a_i(P)>a_{i-1}(P)}\left(1+\frac{z}{t^{a_i(P)}}\right)\mathbf{x}^P\right]$$ $$=\{z^{n-k}\}\left[\sum_{P\in\mathcal{LD}_n}q^{\operatorname{dinv}(P)}t^{\operatorname{area}(P)}\prod_{i\in\operatorname{Val}(P)}\left(1+\frac{z}{q^{d_i(P)+1}}\right)\mathbf{x}^P\right].$$

This conjecture predicts two equivalent labeled-Dyck-path formulas for the Delta-operator expression, relating symmetric-function operators to the combinatorics of area, dinv, and contractible valleys. Its general status is not established by the supplied source context.