The asymptotic $k/2$ conjecture for directed cycle packing and covering

Let $k\ge 3$ be fixed. For a directed graph $D$, let $\nu_k(D)$ be the maximum number of pairwise arc-disjoint directed $k$-cycles, and let $\tau_k(D)$ be the minimum number of arcs whose removal makes $D$ free of directed $k$-cycles.

Asymptotic directed-cycle packing and covering conjecture. For all sufficiently large $n$, every $n$-vertex directed graph $D$ satisfies

$$\tau_k(D)\le (k/2)\nu_k(D).$$

The conjecture proposes improving the paper's dense-case bound with constant $2k/3$ (and $25/8$ for $k=5$) to $k/2$. It is stated only asymptotically in the number of vertices.