A homotopical consequence of branched covers

We prove that the profinite completion of a pseudomanifold is the Artin-Mazur’s etale homotopy type construction on its branched covers, which was implicitly conjectured by Sullivan in his MIT note (page 247) around 1970. This is a consequence of the existence of enough $K(\pi ,1)$ open dense subspaces in a pseudomanifold.