Is every graph whose $k$-vertex subgraphs have at most $2k-3$ edges Ramsey size-linear? (Erdős #566)

For a graph $G$, say $G$ is *Ramsey size-linear* if there is a constant $C=C(G)$ such that $R(G,H)\le C\,m$ for every graph $H$ with $m$ edges and no isolated vertices; here $R(G,H)$ is the least $N$ such that every red/blue colouring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$, and $X\ll Y$ means $X\le CY$ for some constant $C$ independent of $H$. Suppose $G$ has the property that every subgraph of $G$ on $k$ vertices spans at most $2k-3$ edges. Is it then true that $$R(G,H)\ll m$$ for every graph $H$ with $m$ edges and no isolated vertices?

Acceptance. FULLY RESOLVES: a complete proof that every graph $G$ satisfying the subgraph-density condition (every $k$-vertex subgraph has at most $2k-3$ edges) is Ramsey size-linear — machine-checkable (Lean, against the existing formal-conjectures statement) preferred, else a full written proof with all steps; OR a disproof exhibiting a specific graph $G$ meeting the condition together with an explicit infinite family $H_1,H_2,\dots$ (with $m_i$ edges, $m_i\to\infty$) and a proof that $R(G,H_i)/m_i\to\infty$. ADVANCES: enlarge the class of graphs proven Ramsey size-linear strictly beyond the current frontier — currently every $G$ on $n$ vertices with at most $n+1$ edges is known size-linear [EFRS93] — by proving size-linearity for a strictly larger family (e.g. all $G$ on $n$ vertices with at most $n+c$ edges for a fixed $c\ge 2$, or a named graph family satisfying the hypothesis), with proof; OR establish, for a specific sparse $G$ satisfying the hypothesis, an upper or lower bound on $R(G,H)$ (over all $H$ with $m$ edges) strictly sharper than any bound stated in the background, with proof or a reproducible search certificate. Deliver a proof file (Lean or complete written proof), or the counterexample graph plus the divergence proof.