Yes, $(2K_2,K_4)$-free graphs are recolorable

We prove that every $(2K_2,K_4)$-free graph is recolorable. Equivalently, for every such graph $G$ and every $\mathrm{\ell }\ge \chi (G)+1$, the reconfiguration graph of proper $\mathrm{\ell }$-colorings of $G$, in which two colorings are adjacent if they differ on exactly one vertex, is connected. This resolves the final remaining open case in the classification of recolorable $(F_1,F_2)$-free graphs when $F_1$ and $F_2$ have at most four vertices.