Burr–Erdős–Faudree–Rousseau–Schelp conjecture on adding matching components

Let $G$ and $H$ be graphs, and let $K_2$ denote the complete graph on two vertices. A pair $(G,H)$ is Ramsey-finite if it has only finitely many Ramsey-minimal graphs. Burr–Erdős–Faudree–Rousseau–Schelp conjecture. If $(G,H)$ is Ramsey-finite, then $(G\cup \ell K_2,H\cup mK_2)$ is Ramsey-finite for all integers $\ell,m\ge 0$. This conjecture concerns the stability of Ramsey-finiteness under adjoining arbitrary numbers of isolated matching components to the two target graphs. The paper states that this conjecture is true.