Perkins–Perarnau's coloring extremal conjecture for triangle-free and square-free cubic graphs

Let $G$ be a finite graph, let $K_q$ be the complete graph on $q$ vertices, and let $\hom(G,K_q)$ denote the number of graph homomorphisms from $G$ to $K_q$. Perkins–Perarnau's coloring extremal conjecture. (a) Provided that $q\geq 4$, among $3$-regular triangle-free graphs $G$, the quantity

$$\hom(G,K_q)^{1/|V(G)|}$$

is minimized when $G$ is the Petersen graph. (b) Among $3$-regular graphs $G$ without cycles of length $4$, the quantity

$$\hom(G,K_q)^{1/|V(G)|}$$

is maximized when $G$ is the Heawood graph. These assertions propose the extremal cubic graphs for normalized proper $q$-coloring counts under the stated girth restrictions. The first assertion requires $q\geq 4$ because the source gives a counterexample at $q=3$; the status of the two assertions is otherwise left open in the source.