The Dyck-path–tree correspondence for modern intervals and dexter upper ideals

Let $w$ be a Dyck path and let $t=\kappa(w)$ be the corresponding planar rooted binary tree under the bijection $\kappa$ from Dyck paths to planar rooted binary trees. A correspondence conjecture. The number of modern intervals with minimum $t$ in the Tamari lattice equals the size of the dexter upper ideal with minimum $w$. This conjecture predicts an enumerative correspondence between intervals in the Tamari lattice and upper ideals associated with Dyck paths; no resolution is given in the source.