Jones' Conjecture for planar graphs

Let $G$ be a planar graph. A cycle packing is a set of vertex-disjoint cycles in $G$; let $\operatorname{cp}(G)$ denote its maximum size. A feedback vertex set is a set $S$ of vertices such that $G-S$ is a forest; let $\operatorname{fvs}(G)$ denote the minimum size of a feedback vertex set.

Jones' Conjecture. Every planar graph $G$ satisfies

$$\operatorname{fvs}(G)\leqslant 2\cdot\operatorname{cp}(G).$$

This conjecture concerns the relationship between cycle packing and feedback vertex sets, strengthening the general Erdős–Pósa-type bound for planar graphs. The source confirms the conjecture for subcubic planar graphs, but the general planar case remains open.