Zhao's bipartite swapping conjecture

Let $H$ be a graph and let $q\geq 2$. Write $K_q$ for the complete graph on $q$ vertices, and let $H\times K_2$ denote the categorical product with the two-vertex complete graph. Zhao's bipartite swapping conjecture.

$$\hom(H,K_q)^2 \leq \hom(H\times K_2,K_q).$$

This inequality is a proposed bipartite-swapping property for proper $q$-colourings. The paper states that it remains unknown in general, although it is proved for several classes of graphs later in the paper.