Ramsey size-linearity of $Q_3$, $K_{3,3}$, and the subdivided $K_4$: is $R(G,H)\ll m$? (Erdős #567)

A graph $G$ is *Ramsey size-linear* if $R(G,H)\le C(G)\,m$ for every graph $H$ with $m$ edges and no isolated vertices, where $R(G,H)$ is the least $N$ such that every red/blue edge-colouring of $K_N$ yields a red copy of $G$ or a blue copy of $H$ (and $X\ll Y$ means $X\le CY$ for a constant $C$ independent of $H$). Let $G$ be one of the following three graphs: the $3$-cube $Q_3$; the complete bipartite graph $K_{3,3}$; or $H_5$, obtained from the $5$-cycle $C_5$ by adding two vertex-disjoint chords — equivalently $K_4^{*}$, the graph obtained from $K_4$ by subdividing a single edge. Is it true that, for each such $G$, $$R(G,H)\ll m$$ for every graph $H$ with $m$ edges and no isolated vertices?

Acceptance. FULLY RESOLVES: for each of the three graphs $G\in\{Q_3,K_{3,3},H_5\}$, a proof (Lean against the existing formal-conjectures statement preferred, else a complete written proof) that $R(G,H)\le C(G)\,m$ holds for all $H$ with $m$ edges and no isolated vertices, exhibiting an explicit constant $C(G)$; OR, for some such $G$, a disproof giving an explicit infinite family $H_1,H_2,\dots$ with $m_i\to\infty$ and a proof that $R(G,H_i)/m_i\to\infty$. Fully settling any one of the three graphs (proof or disproof of its linearity) resolves that named case. ADVANCES: remove the bipartite restriction in the known bound $R(H_5,H)\ll m$ by extending [BGS23] to a strictly larger class of $H$ (with proof); OR, for $K_{3,3}$ or $Q_3$, prove an upper bound $R(G,H)\ll m\cdot m^{o(1)}$ that improves on any bound stated in the background; OR compute exact values of $R(K_{3,3},H)$ or $R(Q_3,H)$ over a new range of small $H$ with a reproducible search certificate, yielding sharpened constants. Deliver a proof file, a counterexample family with divergence proof, or search code plus certified values.