The polytopal realization conjecture for ornamentation lattices

Let $D$ be a directed graph, and let $\operatorname{\mathcal{O}}(D)$ denote its ornamentation lattice. Polytopal realization conjecture. For any directed graph $D$, the ornamentation lattice

$$\operatorname{\mathcal{O}}(D)$$

is isomorphic to the transitive closure of the graph of a polytope oriented in a linear direction. This conjecture extends the corresponding result for unstarred increasing trees and answers a question posed for rooted trees; it remains open beyond the unstarred-tree case.