The $=5$ case of Vizing's conjecture for triangle-free graphs

Let $G$ be a $5$-regular triangle-free graph.

The $=5$ case of Vizing's conjecture. Every $5$-regular triangle-free graph has chromatic number at most $4$.

This is identified in the source as the current smallest open case of the broader triangle-free graph coloring conjecture. The source notes that the corresponding bound is known for maximum degree at most $4$, but this $5$-regular case was presented as open.