The circular chromatic conjecture for planar graphs

Let $t$ be a positive integer and let $G$ be a planar graph of girth at least $2t$. The circular chromatic conjecture.

$$\chi_c(G)\le 2+\frac{2}{t}.$$

The cases $t=1$ and $t=2$ follow from the Four Color Theorem and Grötzsch's theorem, respectively, while the conjecture remains open for every $t\ge 3$. It is known for $K_4$-minor-free graphs, a subclass of planar graphs.