The distance-packing Erdős–Pósa conjecture for cycles

For a positive integer $d$, a distance-$d$ packing of cycles in a graph $G$ is a set of cycles such that no path of length at most $d$ joins two distinct cycles. For a vertex set $X$, let $B_G(X,r)$ be the vertices at distance at most $r$ from $X$. The distance-packing Erdős–Pósa conjecture. There are functions $f(k)=\mathcal{O}(k\log k)$ and $g(d)=\mathcal{O}(d)$ such that, for all positive integers $k,d$, every graph $G$ contains either a distance-$d$ packing of $k$ cycles or a set $X$ of at most $f(k)$ vertices such that $G-B_G(X,g(d))$ is a forest. This would simultaneously generalize the paper's induced-cycle and distance-$d$ two-cycle results. No resolution is given in the supplied text.