Gromov's generalized Geroch conjecture

Fix a scale $r>0$. For a Riemannian metric on a closed manifold, let $Scal_r(p)$ denote the macroscopic scalar curvature at scale $r$ at $p$, defined by comparing the volume of the radius-$r$ ball in the universal cover with the corresponding ball in the simply connected constant-curvature space. Generalized Geroch conjecture. The $n$-dimensional torus does not admit a metric with

$$Scal_r>0.$$

The source attributes this formulation to Gromov (1985) and describes it as considerably stronger than the original Geroch conjecture and wide open.