The strong Gromov conjecture for classifying maps

Let $M$ be a closed PSC $n$-manifold with torsion-free fundamental group $\pi$, let $B\pi=K(\pi,1)$, and let $u^M:M\to B\pi$ be a classifying map for the universal covering $\widetilde M$. Denote by $B\pi^{(n-2)}$ the $(n-2)$-dimensional skeleton of a CW structure on $B\pi$.

Strong Gromov conjecture. The classifying map can be deformed into the $(n-2)$-skeleton $B\pi^{(n-2)}$.

This is stronger than the macroscopic dimension bound in Gromov's conjecture. The paper notes that the torsion-free restriction is essential, that the statement is false for finite cyclic groups, and that it was known for products of free groups; the general conjecture remains open.