Essentialness conjecture for positive scalar curvature

Let $M$ be a closed oriented manifold, with classifying map $\phi:M\to B\pi_1(M)$ and singular-homology fundamental class $[M]_H$. Call $M$ rationally essential when

$$\phi_*([M]_H)\neq 0\in H_n(B\pi_1(M);\mathbb Q).$$

Essentialness conjecture. An essential manifold does not admit a metric of positive scalar curvature. The paper explains that essential spin$^c$ manifolds are K-theoretically essential, so this conjecture is a homological counterpart of the preceding K-theoretic assertion; its general status is left open.