Outdegree threshold conjecture for directed cycles

An oriented graph is a directed graph without loops or multiple edges. For an oriented graph $G$, let $\delta^+(G)$ denote its minimum outdegree, and let $C_\ell$ denote a directed cycle of length $\ell$. For $\ell\ge4$, let $k$ be the smallest integer greater than $2$ that does not divide $\ell$. Outdegree threshold conjecture. Fix an integer $\ell\ge4$ and let $k$ be the smallest integer greater than $2$ that does not divide $\ell$. There exists $n_0:=n_0(\ell)$ such that every oriented graph $G$ on $n\ge n_0$ vertices with

$$\delta^+(G)\ge \frac{n}{k}+1$$

contains $C_\ell$. The conjecture concerns the gap between outdegree and semidegree thresholds; the paper notes that these thresholds differ by less than one is expected, but does not establish this statement in general.