A (reduced) crossed complex is, intuitively, a non-abelian complex of groups $\ldots \to G_2 \to G_1 \to G_0$ with a $G_0$ action in such that $G_1 \to G_0$ is a crossed module.

There are a couple of constructions that could be called a classifying space of a crossed complex, namely the one given at this MO question, and the simplicial set arising from the $\bar{W}$ functor applied to the simplicial group which is the group T-complex associated to the crossed complex. (Aside: I would be tempted to call this the *groupal nerve*, as opposed to the simplicial set constructed in the process of forming the classifying space.) Then one can apply geometric realisation to get a space.

My question is this:

Does the usual classifying space functor from crossed complexes to topological spaces lift, up to homotopy, through the functor $|\bar{W} - |: sGrp \to Top$?

Edit: An equivalent formulation is this: for $T:Crs \to sGrp$ the functor from crossed complexes to simplicial groups in the above paragraph (this is half of the relevant Dold-Kan correspondence for crossed complexes), $N:Crs \to sSet$ the nerve functor and $\bar{W}:sGrp \to sSet$ the classifying space functor, do we have a (weak) homotopy equivalence $$ \bar{W}T(G) \sim NG? $$