I think the following should be in the literature but couldn't find it.

Recall that around the 1970's Bousfield described the $R$-localization $EX$ of any space $X$, for $R$ a fixed ring. The construction comes with a natural map $X \to EX$, which induces an isomorphism $H_*(X;R) \cong H_*(EX;R)$ and satisfies a universal property.

Then he constructed a model category structure on the category of simplicial sets for which weak equivalences are maps that induce isomorphisms on $H_*( \text{ }; R)$ and the cofibrations are injections.

Later on, Bousfield and Kan, as well as E. Dror Farjoun, described a *fiberwise localization* which, given any pointed space $(X,b)$, may be applied to the fibration $\tilde{X} \to X \to B\pi_1(X,b)$, where $\tilde{X}$ is the universal cover of $X$, to (functorially) produce a new fibration $E\tilde{X} \to E'X \to B\pi_1(X,b)$ for which the fibers are the $R$-localization of $\tilde{X}$.

Question: Is there a model category structure on pointed spaces with the same cofibrations as above and whose weak equivalences are maps $X \to Y$ for which the induced map $E'X \to E'Y$ is a weak homotopy equivalence of fibrations?

I'd appreciate if someone could point out a reference. I couldn't find it explicitly in the work of Bousfield, Kan, or Dror-Farjoun, but it should be in the literature. It should also hold in more general contexts.