The chromatic extremal conjecture for regular graphs

Let $G$ be an $n$-vertex, $d$-regular graph, let $K_q$ be the complete graph on $q$ vertices, and let ${\rm hom}(G,K_q)$ denote the number of graph homomorphisms from $G$ to $K_q$. Chromatic extremal conjecture. For every $n$, $d$ and $q$,

$${\rm hom}(G,K_q)\leq {\rm hom}(K_{d,d},K_q)^{n/2d}.$$

Equivalently, the number of proper $q$-colorings satisfies $c_q(G)\leq c_q(K_{d,d})^{n/2d}$. The conjecture is open for all $q\geq3$; it is known when $q=q(n,d)$ is sufficiently large, but the cited methods do not settle constant $q$.