The two-copy coarse Erdős–Pósa conjecture for planar graphs

Let $H$ be a planar graph. A $K$-fat minor-model of $H$ is a $K$-fat model of $H$, and $B_G(Z,cK)$ denotes the ball of radius $cK$ around $Z$ in $G$. Then there exists a constant $c$ such that for every graph $G$, if $G$ does not contain two $K$-fat minor-models of $H$ at distance at least $K$ from each other, then there is a set $Z$ of at most $c$ vertices of $G$ such that $B_G(Z,cK)$ meets all $K$-fat $H$ minor-models in $G$. Two-copy coarse Erdős–Pósa conjecture.

The paper states this as the unresolved case of two copies after showing that planar graphs do not generally have the coarse Erdős–Pósa property.