A relaxation of the Bermond-Thomassen conjecture

The well-known Bermond-Thomassen conjecture states that every digraph of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint directed cycles. Despite being posed in 1981, this conjecture remains unresolved for all $k \ge 4$. We prove a relaxation of this conjecture: every digraph $D$ of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint cycles, each of which either is directed or can be made directed by reversing one of its arcs. This bound is sharp and answers a question raised by Cames van Batenburg during the online workshop "Entropy Compression and Related Methods" in $2021$.