Chepoi–Estellon–Vaxès linear Erdős–Pósa conjecture for planar ball hypergraphs

Let $G$ be a planar graph and let $u_{\ell}(G)$ and $\tau_{\ell}(G)$ denote, respectively, the packing number and transversality of its hypergraph of balls of radius $\ell$. Chepoi–Estellon–Vaxès' conjecture. There \exists a constant $c$ such that, for every $\ell$ and every planar graph $G$,

$$\tau_{\ell}(G) \leq c \cdot \nu_{\ell}(G).$$

This conjecture asks for a linear gap between the packing and transversal numbers. It is subsumed by the stated result of Dvořák for bounded-expansion classes only with a coefficient depending on $\ell$, so the planar linear bound remains unresolved in the supplied text.