Let $M$ be a model category. I don't assume that $M$ has functorial factorizations or that $M$ is simplicial. Write $M^{c}$ (respectively, $M^{cf}$) for the full subcategory of $M$ on the cofibrant objects (respectively, the cofibrant and fibrant objects).

Does the inclusion $M^{cf} \to M^c$ induce a Dwyer-Kan equivalence on the simplicial localizations at the class of weak equivalences?

Assuming the answer is "yes", does anyone know of a reference? I couldn't find this in the original papers of Dwyer and Kan for general $M$, only when $M$ has functorial factorizations.