Clique-minor separating-pair conjecture

Let $n\in\mathbb{N}$. Clique-minor separating-pair conjecture. There exist non-isomorphic graphs $H$ and $H'$ such that

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

Here $K_n$ is the complete graph on $n$ vertices. The paper says that proving this conjecture for every $n$ would suffice to prove the non-isomorphism conjecture for proper minor- and union-closed families; it is left open.