On the ErdH{o}s-Posa property for long holes in $C_4$-free graphs

We prove that there exists a function $f(k)=mathcal{O}(k^2 log k)$ such that for every $C_4$-free graph $G$ and every $k in mathbb{N}$, $G$ either contains $k$ vertex-disjoint holes of length at least $6$, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least $6$. This answers a question of Kim and Kwon [ErdH{o}s-Posa property of chordless cycles and its applications. JCTB 2020].