Haglund–Remmel–Wilson valley Delta conjecture

Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(q,t)$, let $\{\widetilde H_\mu\}$ be the modified Macdonald basis, and define $\Delta'_f$ by

$$\Delta'_f\widetilde H_\mu=f[B_\mu-1]\widetilde H_\mu,$$

where $B_\mu=\sum_{(i,j)\in\mu}q^{j-1}t^{i-1}$. Let $\operatorname{Val}_{n,k}(x;q,t)$ be the generating function over labelled Dyck paths with $k$ decorated contractible valleys. Valley Delta conjecture. For $0\le k<n$,

$$\Delta'_{e_{n-k-1}}e_n=\operatorname{Val}_{n,k}(x;q,t).$$

The rise analogue is a theorem, while the valley version is open in general. Its symmetry is a necessary consistency property and is itself open in general; specializations and cases including $t=0$, $q=0$, the Schröder case, and low-area cases are known.