Nešetřil's Pentagon Conjecture for high-girth cubic graphs

Let $G$ be a cubic graph and let its girth be the length of its shortest cycle. A graph homomorphism $G\to C_5$ is a vertex map preserving adjacency.

Nešetřil's Pentagon Conjecture. If $G$ is a cubic graph of sufficiently high girth, then $G$ is homomorphic to $C_5$.

Equivalently, sufficiently high-girth cubic graphs should have circular chromatic number at most $5/2$. The statement is known with $C_3$ in place of $C_5$, while replacing $C_5$ by $C_7$ is false; the Pentagon Conjecture itself remains open.