Embedding cycles of given length in oriented graphs

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