Distinguishing conjecture for distinct immersion-closed classes

Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be two distinct immersion-closed and union-closed graph classes. For graphs $G$ and $H$, write $G =_{\mathcal{M}} H$ when $\hom(K,G)=\hom(K,H)$ for every $K\in\mathcal{M}$. Distinguishing conjecture. There exist $i,j\in\{1,2\}$ with $i\ne j$ and graphs $G$ and $H$ such that

$$G =_{\mathcal{M}_i} H \quad\text{and}\quad G \ne_{\mathcal{M}_j} H.$$

This would strengthen the paper's homomorphism-counting results by showing that distinct immersion-closed, union-closed classes induce genuinely different equivalence relations. The source presents it as an open problem, and no resolution is specified.