Kelly's minimum semi-degree conjecture for directed cycles

Let $l\geq 4$ be an integer, and let $k$ be the smallest integer greater than $2$ that does not divide $l$. Let $\delta^0(G)$ denote the minimum semi-degree of an oriented graph $G$. Kelly's conjecture. There exists $n_0$ such that every oriented graph $G$ on $n\geq n_0$ vertices satisfying

$$\delta^0(G)\geq \left\lfloor\frac{n}{k}\right\rfloor+1$$

contains a copy of the directed cycle $C_l$. The conjecture concerns the semi-degree threshold forcing directed cycles of length at least four; the source gives no resolution status.