Odd-girth multicoloring conjecture for hexagonal graphs

Let $k\geq 1$, and let $G$ be a hexagonal graph, meaning a finite triangle-free induced subgraph of the triangular lattice. The odd-girth of $G$ is the length of its shortest odd cycle.

Odd-girth multicoloring conjecture. If $G$ has odd-girth at least $2k+1$, then $G$ is $(2k+1,k)$-colorable.

This conjecture generalizes McDiarmid–Reed's conjecture and is proposed as a possible extension of the paper's results. The source does not provide a proof or a resolution.