Profinite completions and cohomology jump loci

Let $X$ be a connected finite-type CW-complex with fundamental group $G$. We show that the profinite completion $\widehat{G}$ determines the cohomology jump loci $\mathcal{V}^q_s(X,\mathbb{C})$, under two hypotheses treated separately: that the loci are finite unions of torsion-translated subtori, w