Sah–Sawhney–Stoner–Zhao conjecture on antiferromagnetic homomorphism maxima

Let $H$ be a $d$-regular graph, and let $G$ be a graph possibly with loops. Regard $G$ as its symmetric adjacency matrix, and suppose that it has at most one positive eigenvalue. Sah–Sawhney–Stoner–Zhao conjecture. Then

$$\hom(H,G)^{1/v(H)} \leq \hom(K_{d,d},G)^{1/(2d)}.$$

This would generalize the Kahn–Zhao theorem and the corresponding result for complete target graphs, asserting that complete bipartite graphs maximize normalized homomorphism counts from regular graphs under the stated spectral condition. The conjecture is presented as open in the paper.