Keszegh–Pálvölgyi directed-hypergraph 2-coloring conjecture with multiple head-vertices

Prove that every directed hypergraph in which each hyperedge has more tail-vertices than head-vertices, every hyperedge has size at least three, and any two hyperedges intersecting in exactly one vertex have that common vertex as a head-vertex in at least one of the two hyperedges admits a proper 2-coloring, without restricting hyperedges to have exactly one head-vertex.

Background

The paper proves the Keszegh–Pálvölgyi conjecture for directed hypergraphs whose hyperedges have exactly one head-vertex and at least two tail-vertices. The remaining case identified by the authors is when a hyperedge may contain more than one head-vertex, while still having more tail-vertices than head-vertices and satisfying the specified one-vertex intersection condition.

Resolving this problem would establish the full conjecture for directed hypergraphs with hyperedges of size at least three, extending both the paper’s one-headed result and the previously known 3-uniform case.