The $C_4$-factor conjecture for digraphs

Let $\eta>0$. For sufficiently large integers $n$ divisible by $4$, consider an $n$-vertex digraph $G$, where $\delta^0(G)$ denotes its minimum semi-degree.

The $C_4$-factor conjecture. There exists $n_0=n_0(\eta)\in\mathbb N$ such that, for every $n\geq n_0$ divisible by $4$, if

$$\delta^0(G)\geq (1/2+\eta)n,$$

then $G$ contains a $C_4$-factor. This is presented as an open problem concerning even cycle factors, following the asymptotic determination of the corresponding threshold for orientations of odd cycles; the even-cycle case is described as more challenging.