Erdős–Pósa conjecture for long cycles through prescribed vertices

Let $G$ be a graph, let $S\subseteq V(G)$, and let $k,\ell$ be positive integers. An $S$-cycle is a cycle containing a vertex of $S$.

Long $S$-cycle Erdős–Pósa conjecture. For every graph $G$, every subset of vertices $S$, and every pair of positive integers $k,\ell$, there is either a set of $k$ disjoint $S$-cycles of length at least $\ell$, or a set $X$ with

$$|X|=O\bigl(k(\ell+\log k)\bigr)$$

such that $G-X$ contains no $S$-cycle of length at least $\ell$.

This conjecture improves the known bound of order $O(\ell k\log k)$ for the corresponding hitting set when both parameters vary. The examples discussed in the source show that the dependence on $k$ and on $\ell$ cannot generally be reduced when the other parameter is fixed, while the optimal bound when both grow remains open.