Erdős–Pósa-type conjecture for cycles of distinct lengths

For a graph $G$, let $\Lambda(G)=\{\ell\in\mathbb{N}\mid G\text{ contains a cycle of length }\ell\}$, and let $G-X$ denote the graph obtained by deleting the vertices in $X$. Distinct-length cycle packing-cover conjecture. There exists a function $f(k)\in\mathcal{O}(k\log k)$ such that every graph $G$ contains $k$ vertex-disjoint cycles with distinct lengths, or a set $X$ of at most $f(k)$ vertices such that

$$|\Lambda(G-X)|\leq k-1.$$

This proposes an Erdős–Pósa-type dual certificate for cycles of pairwise distinct lengths; the source presents it as a possible improvement and does not state that it has been resolved.