Dranishnikov's rational inessentiality conjecture for macroscopic smallness

Let $M$ be a closed oriented $n$-dimensional manifold, let $\pi=\pi_1(M)$, and let $B\pi$ be a classifying space. If $f\colon M\to B\pi$ classifies the universal bundle, call $M$ rationally inessential when

$$f_*([M])=0\in H_n(B\pi;\mathbf Q).$$

Call $M$ macroscopically small when it is not macroscopically large, where macroscopic largeness means $\dim_{mc}(\widetilde M)=n$. Dranishnikov's rational inessentiality conjecture. If $M$ is rationally inessential, then it is macroscopically small. The conjecture gives a homological criterion for failure of macroscopic largeness. The surrounding discussion attributes this formulation to Dranishnikov; the supplied text does not state a general resolution.