The site labels the problem OPEN, and no claim about it exists, so the frontmatter standing is open. No source cited here decides any of the three cases. The one accepted formal result, a Lean refutation accepted by the bounty site Conjectures.io on 7 August 2026 (record 145e01a5-a004-4b17-a9fa-7503a28ff052), refutes a defective formalization of the $Q_3$ case in which the size Ramsey number $\hat r(Q_3,H)$ stood in for $R(Q_3,H)$; the site classifies it as a formalization-defect award that does not settle the problem, and it bears on none of the three cases (see Formalization). The partial results are Theorem 3 (Bradač, Gishboliner and Sudakov 2022) of [BGS24], $R(K_4^*,F)=O(e(F))$ for every bipartite $F$ without isolated vertices, and its Theorem 4 (Bradač, Gishboliner and Sudakov 2022), Ramsey size linearity for each subdivision of $K_4$ having six or more vertices, which leaves out the five-vertex $K_4^*=H_5$; from [EFRS93], Theorem 5 (Erdős et al. 1993) covers the one-edge-deleted graphs $K_{3,3}-e$ and $Q_3-e$ (p. 395), and Corollary 1 (Erdős et al. 1993) is why $K_4$ itself fails. For the complete-graph target, Theorem 2 (Bradač, Gishboliner and Sudakov 2022) of [BGS24] gives $R(G,K_n)=O(n^3)$ for each of the three graphs (each is connected with $e(G)-v(G)\le4$; for $H_5=K_4^*$ its Section 6 proves $O(n^{5/2})$), where Ramsey size-linearity would need $O(n^2)$, and Theorem 2 of [EFRS93] gives the lower bound $R(G,K_n)>C(n/\log n)^{(q-1)/(p-2)}$ with exponent $2$ for $K_{3,3}$ and $H_5$ and $11/6$ for $Q_3$ (specializations made here). This is a bounded negative finding from the searches, not a certificate of openness.