  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 [Erd\H{o}s-P\'osa property of chordless
cycles and its applications. JCTB 2020].
