Assume that we have a connective spectrum $X$, and denote the $p$-completion of this spectrum in the sense of Bousfield by $X^{\wedge}_p$ (which is given by the function spectrum $F(S^{-1}\mathbb{Z}/p^{\infty},X)$). Let $(\Omega^{\infty}_0X)^{\wedge}_p$ be the profinite $p$-completion of the zeroth component of the infinite loop space of $X$. Is that true that $\Omega^{\infty}_0X^{\wedge}_p\simeq (\Omega^{\infty}_0X)^{\wedge}_p$, meaning that they are weakly equivalent as topological spaces ?

Remark. I was thinking about two specific spectra $X$ and $Y$ and a map between them which I spare you with their descriptions that I know the map induces equivalence between $p$-completions $X^{\wedge}_p\simeq Y^{\wedge}_p$. I'd like to show $H_*(\Omega^{\infty}_0X,\mathbb{Z}/p)\cong H_*(\Omega^{\infty}_0Y,\mathbb{Z}/p)$. This made me to ask the previous question.