Cranston–Rabern's chromatic bound conjecture for vertex-transitive graphs

Let $G$ be a vertex-transitive graph, with chromatic number $\chi(G)$, clique number $\omega(G)$, and maximum degree $\Delta(G)$. Cranston–Rabern's conjecture. Every vertex-transitive graph satisfies

\chi(G) \leq \max \left\\{ \omega(G), \left\lceil\frac{5\Delta(G) + 3}{6}\right\rceil \right\\}.

The conjecture is supported by the bounds and cases proved in the paper; its status is not resolved in the supplied source.