Minor-separating graph-pair conjecture

Let $G$ be a connected graph, and let $H$ and $H'$ be graphs. Write $\hom(F,H)$ for the number of homomorphisms from $F$ to $H$. Minor-separating graph-pair conjecture. For every connected graph $G$, there exist graphs $H$ and $H'$ such that

$$\hom(G,H)\ne\hom(G,H')$$

and

$$\hom(F,H)\ne\hom(F,H')\ \Longrightarrow\ F\text{ contains }G\text{ as a minor}.$$

Equivalently, if $\mathcal{F}_G$ is the family of graphs not containing $G$ as a minor, then $H\cong_{\mathcal{F}_G}H'$ but the homomorphism counts from $G$ differ. The paper describes this as one of four equivalent open conjectures and regards it as the most approachable.