I believe that the following is true, but I cannot find a proof. Let $X_\bullet$ be a simplicial topological space (I can add that my $X_\bullet$ comes from a bisimplicial set, so the spaces $X_n$ are CW-complexes). Suppose that all the face and degeneracies maps are homotopy equivalences. Is it true that the geometric realization $|X_\bullet|$ is homotopy equivalent to the space $X_0$? (Possibly under some mild extra-hypotheses, e.g. connectedness of the $X_n$, etc.?)
Thanks in advance!