Dvořák–Mnich conjecture on fractional coloring of planar graphs of girth five

Let $G$ be a planar graph of girth at least five, and let $\chi_f(G)$ denote its fractional chromatic number. Dvořák–Mnich conjecture. There exists a real number $c<3$ such that every planar graph of girth at least five satisfies

$$\chi_f(G)\leq c.$$

This is the girth-five special case identified as a key step toward the broader conjecture for triangle-free plane graphs without separating $4$-cycles. It remains open in the supplied text; the analogous bounded-away-from-$3$ result is known under maximum degree at most $4$.