Directed regular graph cycle-cover conjecture

Let $G$ be a directed $d$-regular graph on $n$ vertices, and regard a cycle as a directed cycle, with an individual edge allowed to count as a cycle of length two as in the source. Directed cycle-cover conjecture. Every directed $d$-regular $n$-vertex graph has a collection of $\mathcal O(n/d)$ vertex-disjoint cycles that cover all vertices of the graph. This is presented as a potential generalisation of the Feige–Fuchs conjecture; it remains open.