Let $\mathrm{Top}$ be the model category of topological spaces. Define a new model structure on $\mathrm{Top}$ where $f:X\rightarrow Y$ is a weak equivalence iff $$f_{\ast}:H_{\ast}(X,\mathbb{F}_{p})\rightarrow H_{\ast}(Y,\mathbb{F}_{p})$$ is an isomorphism. Let $X \mapsto L(X)$ be a functorial fibrant replacement in the new model category.

**Questions:**

If $X\rightarrow Y$ is an $E_{\infty}$-map between $E_{\infty}$-spaces, is it true that $L(X)\rightarrow L(Y)$ is an $E_{\infty}$-map between $E_{\infty}$-spaces?

If $X$ is a pointed connected space (simply connected?), is it true that $$ L(\Omega X)\sim \Omega L(X) \, ? $$

If $X$ is a connected $E_{\infty}$-space, is it true that $$ L(BX)\sim B(LX) $$ as $E_{\infty}$-spaces, where $B$ is the bar construction?