Universal bipartite-swapping conjecture for antiferromagnetic graphs

Let $G$ be an antiferromagnetic graph, and let $H$ be any graph. Universal bipartite-swapping conjecture. Then

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

Here $H\times K_2$ is the categorical product with the two-vertex complete graph. The conjecture would unify the paper's bipartite-swapping results and implies both the Zhao conjecture and the Sah–Sawhney–Stoner–Zhao conjecture. It is presented as an open question in the concluding remarks.