Gromov's macroscopic dimension conjecture for positive scalar curvature

Let $(M^n,g)$ be a complete non-compact manifold with scalar curvature

$$Sc(g) \geq n(n-1).$$

The macroscopic dimension of $M$, denoted by $\operatorname{macrodim}(M)$, is the least integer $k$ for which the large-scale geometry of $M$ admits a uniformly cobounded map to a $k$-dimensional simplicial complex.

Gromov's conjecture. The macroscopic dimension of $M$ satisfies

$$\operatorname{macrodim}(M) \leq n-2.$$

This conjecture predicts that positive scalar curvature controls the large-scale size of a complete non-compact manifold in codimension at least two. The source notes that it remains open even for the weaker bound $\operatorname{macrodim}(M) \leq n-1$.