Let $G,H$ be infinite simple undirected graphs with the property that for any graph $X$ we have $|\text{Hom}(X,G)| = |\text{Hom}(X,H)|$. Does this imply that $G$ is isomorphic to a subgraph of $H$, and vice versa?

(Note that in the finite case the condition above implies $G\cong H$, see here.)