On Gromov's conjecture for totally non-spin manifolds

Gromov's Conjecture states that for a closed $n$-manifold $M$ with positive scalar curvature the macroscopic dimension of its universal covering $\tilde M$ satisfies the inequality $\dim_{mc}\tilde M\le n-2$\cite{G2}. We prove this inequality for totally non-spin $n$-manifolds whose fundamental group is a virtual duality group with $vcd\ne n$. In the case of virtually abelian groups we reduce Gromov's Conjecture for totally non-spin manifolds to the vanishing problem whether $H_n(T^n)^+= 0$ for the $n$-torus $T^n$ where $H_n(T^n)^+\subset H_n(T^n)$ is the subgroup of homology classes which can be realized by manifolds with positive scalar curvature.