Mapping spaces of Swiss cheese operads

We show that the color restriction map ${\mathrm{O}\mathrm{p}}^h({\mathcal{S}\mathcal{C}}_m,{\mathcal{S}\mathcal{C}}_n)\to {\mathrm{O}\mathrm{p}}^h({\mathcal{E}}_{m-1},{\mathcal{E}}_{n-1})$ from the derived mapping space of Swiss cheese operads to that of little discs operads, is a weak homotopy equivalence. We explain how this can help in the study of disc concordance embedding spaces.