Lichiardopol's minimum out-degree conjecture for directed cycles of distinct lengths

For a digraph $D$, its minimum out-degree is the minimum number of outgoing edges over all vertices of $D$. Lichiardopol's conjecture. For every $k\geq1$, there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of distinct lengths. The source states that this directed analogue is famously open and is not known for any minimum out-degree condition depending only on $k$.