The fractional chromatic conjecture for 4-cycle-free planar graphs

Let $G$ be a 4-cycle-free planar graph, and let $\chi_f(G)$ denote its fractional chromatic number. Fractional chromatic conjecture. There exists some $\epsilon>0$ such that

$$\chi_f(G)\leq 4-\epsilon.$$

The conjecture asks for a uniform improvement over the Four Color Theorem for the fractional chromatic number of 4-cycle-free planar graphs. The supplied text presents it as an open question, and no resolution is given in the status evidence.