Homomorphism-distinguishing closure conjecture for minor- and union-closed families

Let $\mathcal{F}$ be a family of graphs. Call it homomorphism distinguishing closed if, for every graph $F\notin\mathcal{F}$, there exist graphs $G$ and $H$ such that $G\cong_{\mathcal{F}}H$ but $\hom(F,G)\ne\hom(F,H)$. Homomorphism-distinguishing closure conjecture. Every minor- and union-closed class of graphs is homomorphism distinguishing closed. The paper states that this conjecture is equivalent to the preceding conjectures and remains open.