Half-integral Erdős–Pósa conjecture for directed cycles of distinct lengths

For a digraph $D$, a set of directed cycles has distinct lengths when no two of its cycles have the same length. Half-integral distinct-length directed-cycle conjecture. For every $k\geq1$, there exists an integer $g(k)$ such that every digraph $D$ contains a set $\mathcal{C}$ of $k$ directed cycles of different lengths such that there is no vertex $v\in V(D)$ contained in three distinct cycles of $\mathcal{C}$, or a set $X$ of at most $g(k)$ vertices such that $D-X$ contains at most $g(k)$ different directed cycle lengths. The source leaves this half-integral variant as an open problem after describing a counterexample to the corresponding vertex-disjoint directed-cycle statement.