Distinctness conjecture for minor- and union-closed graph families

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two distinct minor- and union-closed families of graphs. For a graph family $\mathcal{F}$, write $G\cong_{\mathcal{F}}H$ when $G$ and $H$ have the same number of homomorphisms from every graph in $\mathcal{F}$. Distinctness conjecture. The relations $\cong_{\mathcal{F}_1}$ and $\cong_{\mathcal{F}_2}$ are distinct. This is the paper's main conjecture and is motivated by the question of whether minor- and union-closed families yield different homomorphism indistinguishability relations; its status is presented as open.