Thomassé's conjecture for oriented graphs

Let $D$ be an oriented graph, and let $\delta$ denote its minimum out-degree. Thomassé's conjecture. The graph $D$ contains a directed path of length $2\delta$.

This is the $g=3$ case of Thomassé's proposed strengthening of the Caccetta–Häggkvist conjecture. The general conjecture is false for every $g\geq 4$, while this oriented-graph case remains open.