Galvin's biregular homomorphism conjecture

Let $G$ be a bipartite graph, let $d_u$ denote the degree of $u\in V(G)$, and let $H$ be a loop-graph. Write $\hom(G,H)$ for the number of graph homomorphisms from $G$ to $H$. Galvin's biregular homomorphism conjecture.

$$\hom(G,H)\le \prod_{uv\in E(G)}\hom(K_{d_u,d_v},H)^{1/(d_ud_v)}.$$

This would extend the regular and biregular homomorphism inequalities, as well as the bipartite case of Kahn's irregular independent-set conjecture. Its status is open.