Matherne–Morales–Selover conjecture on unit interval orders and the zeta bijection

Let $\mathcal U_n$ be the set of unit interval orders on $n$ elements, and let $\mathcal D_n$ be the set of Dyck paths from $(0,0)$ to $(n,n)$. For $U\in\mathcal U_n$, let $a(U)$ be the Dyck path whose area set consists of the boxes $(i,j)$ such that $i<j$ and $i\not\prec j$. Let $p(U)$ be the Dyck path whose area sequence is the unique area-sequence part listing of a poset isomorphic to $U$, and let $\zeta$ denote Haglund's zeta bijection on Dyck paths. Matherne–Morales–Selover conjecture. For every $U\in\mathcal U_n$, one has

$$a(U)=\zeta(p(U)).$$

The conjecture identifies the two natural bijections from unit interval orders to Dyck paths: the incomparability-area construction and the part-listing construction. It was proposed on the basis of computer evidence; the paper proves it, so the conjecture is now solved.