The circular homomorphism conjecture for planar graphs of girth at least $4k$

Let $k$ be a positive integer and let $G$ be a planar graph whose girth is at least $4k$. A homomorphism from $G$ to $C_{2k+1}$ is a map preserving adjacency.

Circular homomorphism conjecture. Every planar graph $G$ of girth at least $4k$ admits a homomorphism to $C_{2k+1}$. Equivalently,

$$\chi_c(G) \leq \frac{2k+1}{k}.$$

This is a well-known conjecture giving a homomorphism bound for planar graphs of large girth; its resolution status is not specified in the source.