Galvin–Tetali conjecture on colorings of regular graphs

Let $c_q(G)$ denote the number of proper $q$-colorings of a graph $G$. Let $G$ be an $n$-vertex $d$-regular graph, with $q\ge 3$, and let $K_{d,d}$ be the complete bipartite graph with parts of size $d$. Galvin–Tetali conjecture. The number of proper $q$-colorings satisfies

$$c_q(G) \le c_q(K_{d,d})^{n/(2d)}.$$

Galvin and Tetali proved the inequality for bipartite $G$; the supplied text records proofs for the cases $d=3$ and $d=4$, while the conjecture remains open in general.