Positive scalar curvature and strongly inessential manifolds

We prove that a closed $n$-manifold $M$ with positive scalar curvature and abelian fundamental group admits a finite covering $M^{\mathrm{\prime }}$ which is strongly inessential. The latter means that a classifying map $u:M^{\mathrm{\prime }}\to K({\pi }_1(M^{\mathrm{\prime }}),1)$ can be deformed to the $(n-2)$-skeleton. This is proven for all $n$-manifolds with the exception of 4-manifolds with spin universal coverings.