Burr's stronger finite-characterization conjecture for Ramsey-minimal graph families

Let $G$ and $H$ be graphs, and let $\mathcal{R}(G,H)$ denote the family of Ramsey-minimal graphs for the pair $(G,H)$. A matching is a graph whose connected components are all isomorphic to $K_2$; $S(r)$ denotes the star with $r$ leaves. Burr's conjecture. The family $\mathcal{R}(G,H)$ is finite if and only if either one of $G$ and $H$ is a matching, or both $G$ and $H$ are of the form $S(r)\cup qK_2$ with $r$ a positive odd integer and $q\ge 0$. This proposed characterization is stronger than the two 1981 one-sided statements. It is false, as indicated by the paper's status evidence that Burr's stronger survey conjecture is disproved.