Source characterization of the hypergraphic posets

For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_{\mathbb{H}}$ is the transitive closure of the oriented $1$-skeleton of the hypergraphic polytope ${\mathrm{\Delta }}_{\mathbb{H}}$. In a recent paper, N. Bergeron and V. Pilaud provided a characterization of $P_{\mathbb{H}}$ based on the sources of acyclic orientations for interval hypergraphs. The goal of this work is to extend this source characterization of $P_{\mathbb{H}}$ for arbitrary hypergraphs on $[n]$.