The girth-five fractional-coloring gap conjecture for planar graphs

A planar graph of girth at least five is a planar graph containing no cycle of length less than five. Its fractional chromatic number is denoted by $\chi_f$.

Girth-five fractional-coloring conjecture. There exists $\delta>0$ such that every planar graph of girth at least five has fractional chromatic number at most

$$3-\delta.$$

The paper presents this as a seemingly simpler variant of the conjecture for triangle-free planar graphs without separating 4-cycles and suggests that the two formulations are likely equivalent. The conjecture is not resolved in the source.