Gromov's rational essentialness conjecture for positive scalar curvature

Let $M$ be a closed $n$-manifold, with classifying map $f:M\to K(\pi_1(M),1)$. It is $\mathbb{Q}$-essential if

$$f_*[M]\neq 0\in H_n(K(\pi_1(M),1);\mathbb{Q}).$$

Gromov's rational essentialness conjecture. A closed $\mathbb{Q}$-essential $n$-manifold does not admit any Riemannian metric with positive scalar curvature.

This strengthens the aspherical-manifold conjecture, since every closed aspherical manifold is $\mathbb{Q}$-essential. The source attributes the stronger conjecture to Gromov and gives no resolution.