The cycle-factor conjecture for dense digraphs

Let $D$ be a digraph of order $n$, with minimum semi-degree $\delta^0(D)$. Let $n=n_1+\cdots+n_k$ be a positive integer partition with $n_i\geq 3$ for each $i$. A cycle factor of type $(n_1,\ldots,n_k)$ consists of $k$ vertex-disjoint cycles of lengths $n_1,\ldots,n_k$. Cycle-factor conjecture. Every digraph of order $n$ with

$$\delta^0(D)\geq \frac{2n}{3}$$

contains all possible cycle factors; that is, for every such partition, $D$ has $k$ disjoint cycles of lengths $n_1,\ldots,n_k$, respectively. The source notes that the bound is asymptotically correct when all parts have size $3$ or all parts have size $4$, but gives no resolution of the full assertion.