The 2n/3 minimum semi-degree conjecture for directed W-cycle-factors

Let $D$ be a digraph of order $n$, let $W\subseteq V(D)$, and let

$$|W|=n_1+\cdots+n_k,$$

where each $n_i\geq 3$. An arbitrary $W$-cycle-factor is a collection of $k$ disjoint directed cycles $C_1,\ldots,C_k$ such that $|V(C_i)\cap W|=n_i$ for every $i$.

The $2n/3$ minimum semi-degree conjecture. If

$$\delta^0(W)\geq 2n/3,$$

then $D$ contains an arbitrary $W$-cycle-factor.

The conjecture improves the paper's proved $(3n-3)/4$ minimum semi-degree condition when all prescribed parts have size at least $3$. The source gives a sharpness example below the proposed threshold but does not report a resolution.