Find additional non-singleton equivalence classes

Establish whether the preorder on connected graphs induced by rainbow forbidden subgraphs has any non-singleton equivalence class, other than the classes $\{K_{1,k},K^+_{1,k}\}$ for $k\ge 3$.

Background

The paper defines an equivalence relation $H_1\equiv H_2$ by requiring both $H_1\le H_2$ and $H_2\le H_1$, and studies the resulting partial order on equivalence classes. It proves that, for every $k\ge 3$, the equivalence class containing the star $K_{1,k}$ is exactly $\{K_{1,k},K^+_{1,k}\}$.

The authors state that they have not identified any other non-singleton equivalence classes and pose the question of whether such classes exist elsewhere in the partially ordered set.