n-connective version of Quillen theorem A

Question

sorry for my limited knowledge of homotopy theory, the following question might actually be trivial. Let $F:\mathcal{C} \rightarrow \mathcal{D}$ be a functor, Quillen's theorem A claims if the slice categories $F\downarrow d$ for $d\in \mathcal{D}$ are contractible then $F$ induces a homotopy equivalence. Is the following modification true: Instead of assuming that the slice categories are contractible let's assume their first $n$ homotopy groups vanish. Then does it imply that $F$ induces isomorphism on the first $n$ homotopy groups? (I think the problem boils down to the question that if we have two diagrams of topological spaces and a map between two diagrams that induces isomorphims on the first $n$ homotopy groups, then does the induced map on the homotopy colimit also induce isomorphism on the first $n$ homotopy groups?)