Rosenberg–Weinberger positive scalar curvature conjecture for non-spin universal covers

Let $M$ be a compact oriented manifold whose universal covering does not admit a spin structure. Write $\Gamma=\pi_1(M)$, let $n=\dim M\ge 5$, and let

$$f\colon M\longrightarrow \underline{B}\Gamma$$

be the composition of the classifying map

$$c\colon M\longrightarrow B\Gamma$$

of the universal covering with the natural map $B\Gamma\longrightarrow\underline{B}\Gamma$. Denote by $[M]$ the fundamental class of $M$ in $H_n(M)$. Rosenberg–Weinberger conjecture. The manifold $M$ admits a metric of positive scalar curvature if and only if

$$f_*[M]=0\quad\text{in }H_n(\underline{B}\Gamma).$$

This conjecture proposes a homological characterization of positive scalar curvature for manifolds with non-spin universal covers. The paper presents a counterexample, based on a counterexample to the unstable Gromov–Lawson–Rosenberg conjecture, so the stated equivalence is refuted.