Inclusion conjecture for homomorphism indistinguishability relations

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be minor- and union-closed families of graphs. Write $\cong_{\mathcal{F}_1}\Rightarrow\cong_{\mathcal{F}_2}$ when the first homomorphism indistinguishability relation implies the second for all graphs. Inclusion conjecture.

$$\cong_{\mathcal{F}_1}\Rightarrow\cong_{\mathcal{F}_2}\quad\Longleftrightarrow\quad\mathcal{F}_1\supseteq\mathcal{F}_2.$$

One direction is immediate from inclusion of the testing families, and the conjecture asserts the converse for minor- and union-closed families. The paper states that this conjecture is open and equivalent to the distinctness conjecture.