Unresolved Ramsey-equivalence question for connected graphs

Determine whether there exist non-isomorphic connected graphs H_1 and H_2 that are Ramsey-equivalent, meaning that \mathcal{M}_2(H_1)=\mathcal{M}_2(H_2).

Background

The paper recalls an unresolved question from the Ramsey-equivalence literature concerning non-isomorphic connected graphs with identical families of minimal 2-Ramsey graphs. This question is distinct from the paper’s subsequently solved Maker–Breaker analogue.