Fractional Kowalik–Luzar–Škrekovski conjecture for high-girth planar graphs

Let $G$ be a planar graph of girth at least five, and let $\fva(G)$ denote its fractional vertex-arboricity. Fractional Kowalik–Lu\v{z}ar–Škrekovski conjecture. Every planar graph of girth at least five has fractional vertex-arboricity at most $10/7$, that is,

$$\fva(G)\leq \frac{10}{7}.$$

This is the fractional analogue of the conjecture that such a graph has an induced forest on at least seven-tenths of its vertices. The paper records a best known induced-forest bound of $2n/3$ for planar graphs of girth at least five and leaves the proposed fractional bound open.