The long-cycle induced Erdős–Pósa conjecture

For a graph $G$, an induced packing of cycles is a collection of cycles with no edge between distinct cycles. For a vertex set $X$, let $B_G(X,1)$ be its closed distance-one neighborhood. The long-cycle induced Erdős–Pósa conjecture. There exists a function $f(k,\ell)=\mathcal{O}(k\ell+k\log k)$ such that, for all integers $k\geq1$ and $\ell\geq3$, every graph $G$ contains either an induced packing of $k$ cycles of length at least $\ell$, or a set $X$ of at most $f(k,\ell)$ vertices such that $G-B_G(X,1)$ has no cycle of length at least $\ell$. The conjecture is known when $\ell$ is constant, and the source explains that the order $k\ell+k\log k$ is best possible up to a multiplicative constant; the variable-$\ell$ case remains open.