Refined macroscopic dimension conjecture for the Riemann invariant

Let $M$ be a compact manifold of dimension $n$, let ${\rm \dim}_{mc}(M)$ denote the macroscopic dimension of its universal cover, and let $d$ be an integer satisfying

$$1\leq d\leq n-1.$$

Refined macroscopic dimension conjecture. The inequality

$${\mathbf {Riem}}(M)>\frac{(n-d)(n-d-1)}{2}$$

should imply

$${\rm \dim}_{mc}(M)<d.$$

This conjecture refines the proposed relationship between the ${\mathbf {Riem}}$ invariant and the macroscopic dimension of the universal cover. The supplied text does not state whether the conjecture has been proved or disproved.