Kowalik–Lużar–Skrekovski conjecture for planar graphs of girth at least five

A feedback vertex set of a graph $G$ is a set $S\subseteq V(G)$ such that $G-S$ is a forest. Let $\phi(G)$ denote the minimum size of a feedback vertex set of $G$. The girth of a graph is the length of its shortest cycle, with girth infinity for an acyclic graph. Kowalik–Lużar–Skrekovski conjecture. If $G$ is a planar graph of girth at least five on $n$ vertices, then

$$\phi(G)\leq\frac{3n}{10}.$$

The bound is tight because the dodecahedron attains it. The supplied text does not state a resolution of this conjecture; the source paper presents it as an active problem and develops improved bounds toward it.