OPEN

Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices,
\[R(G,H)\ll m?\]

In other words, is $G$ Ramsey size linear?