On Gromov's scalar curvature conjecture

We prove the Gromov conjecture on the macroscopic dimension of the universal covering of a closed spin manifold with a positive scalar curvature under the following assumptions on the fundamental group: 1. The Strong Novikov Conjecture holds for $\pi$. 2. The natural map $per:ko_n(B\pi)\to KO_n(B\pi)$ is injective.