Thomassé's path-length conjecture for oriented graphs

An oriented graph is a digraph without 2-cycles. Let $h$ and $g$ be positive integers, and let $l(h,g)$ be the minimum integer such that every oriented graph of girth $g$ and minimum outdegree at least $h$ contains a directed path of length $l(h,g)$. Thomassé's conjecture.

$$l(h,g)\geq h(g-1).$$

Sullivan attributed this conjecture to Thomassé and noted that it would imply the Caccetta–Häggkvist conjecture for oriented graphs. The paper disproves its even-girth case, so the conjecture as stated is refuted.