Embedding cycles of given length in oriented graphs

Kelly, Kühn and Osthus conjectured that for any ?≥4$?\ge 4$ and the smallest number k≥3$k\ge 3$ that does not divide ?$?$, any large enough oriented graph G$G$ with δ⁺(G),δ⁻(G)≥⌊|V(G)|/k⌋+1${\delta }^{+}\left(G\right),{\delta }^{-}\left(G\right)\ge \lfloor \left|V\left(G\right)\right|/k\rfloor +1$ contains a directed cycle of length ?$?$. We prove this conjecture asymptotically for the case when ?$?$ is large enough compared to k$k$ and k≥7$k\ge 7$. The case when k≤6$k\le 6$ was already settled asymptotically by Kelly, Kühn and Osthus.