Are there infinitely many graphs $G$ which are not Ramsey size linear but such that all of its subgraphs are?

OPEN

We say $G$ is Ramsey size linear if $R(G,H)\ll m$ for all graphs $H$ with $m$ edges and no isolated vertices.

Are there infinitely many graphs $G$ which are not Ramsey size linear but such that all of its subgraphs are?

Asked by Erdős, Faudree, Rousseau, and Schelp [EFRS93]. $K_4$ is the only known example of such a graph.