Classifying-map conjecture for principal simplicial group bundles

Let $p:E\to B$ be a principal $K$-bundle, where $K$ is a simplicial group, and let $\alpha$ be an atlas for $p$. The atlas induces a functor $\alpha_*:\int B\to\int K$ and hence a map on nerves

$$N\alpha_*:N\int B\to N\int K.$$

Classifying-map conjecture. Under the relevant zigzag of equivalences, the map $N\alpha_*$ is a classifying map for $p:E\to B$: its homotopy class coincides with $[\hat r]\in[B,\overline{W}K]$.

This conjecture is intended to identify the nerve of the transition functor determined by an atlas with the usual classifying map of the principal bundle. The supplied text does not state whether it is known or unresolved.