The planar high-girth circular chromatic number conjecture

Let $G$ be a planar graph with girth $f(g)$.

Circular chromatic number conjecture. If $f(g) \geq 4g$, then the circular chromatic number of $G$ is at most $2+\frac{1}{g}$.

This is presented as an important conjecture in the theory of nowhere-zero flows and as a relaxation of Jaeger's Conjecture for planar graphs, via the equivalent dual formulation using circular chromatic number. Its status is not resolved in the supplied text.