Keszegh–Pálvölgyi conjecture on 2-coloring one-headed directed hypergraphs

Let $H=(V,E)$ be a directed hypergraph, with each hyperedge partitioned into head-vertices and tail-vertices; write $h(e)$ and $t(e)$ for the head and tail of a hyperedge $e$. A proper 2-coloring is a coloring of the vertices with two colors such that every hyperedge contains vertices from at least two color classes.

Keszegh–Pálvölgyi conjecture. If every hyperedge has more tail-vertices than head-vertices, and for every $e_{1},e_{2}\in E(H)$ with $|e_{1}\cap e_{2}|=1$ the common vertex is a head-vertex of at least one of $e_{1},e_{2}$, then $H$ admits a proper 2-coloring.

This conjecture generalizes Lovász's sufficient condition for proper 2-colorability of hypergraphs. The source states that it is proved in the special case where every hyperedge has exactly one head-vertex; in that case the size condition means that every hyperedge has at least two tail-vertices. The general conjecture remains unresolved in the supplied text.