Weak finiteness conjecture for prescribed numbers of different cycle lengths

Let $h(k,\ell)$ be the source's minimum-outdegree threshold for forcing $k$ vertex-disjoint directed cycles with $\ell$ different lengths. Weak finiteness conjecture. For every positive integer $\ell$, if the positive integer $k$ is sufficiently large compared with $\ell$, then

$$h(k,\ell)\text{ is finite}.$$

This is proposed as a weaker alternative to Lichiardopol's conjecture on the finiteness of $h(k,k)$. The source gives no resolution and presents the statement for further research.