Roberson's homomorphism-counting conjecture for proper minor-closed classes

Let $\mathcal{M}$ be a graph class that is proper, minor-closed, and union-closed. For graphs $G$ and $H$, write $G =_{\mathcal{M}} H$ when $\hom(K,G)=\hom(K,H)$ for every $K\in\mathcal{M}$. Roberson's conjecture. If $\mathcal{M}$ is a proper, minor-closed, and union-closed graph class, then there are non-isomorphic graphs $G$ and $H$ such that

$$G =_{\mathcal{M}} H.$$

This conjecture says that homomorphism counts from any such proper minor-closed class do not characterize graphs up to isomorphism. The source attributes it to Roberson; its resolution is not specified here.