Let $X$ be a projective variety (over $\mathbb{C}$). Is there a simplicial object $X_{\bullet}$ in the category of smooth projective varieties and a morphism of simplicial varieties $X_{\bullet} \rightarrow X$ inducing a weak homotopy equivalence (of spaces) $|X_{\bullet}| \rightarrow X$?

This (I believe) is claimed by Donu Arapura in a comment on the following question: Deligne's Mixed Hodge Theory

The statements I've seen about such resolutions (e.g. in Deligne's *Hodge III*) are all assertions about sheaf cohomology, and not homotopy. I'd be happy with just a reference, or an argument that this is a direct consequence of the statements I have alluded to.