Higher-dimensional Urysohn-width conjecture under positive scalar curvature

Determine whether a uniform positive scalar-curvature lower bound on an n-dimensional Riemannian manifold controls its (n−2)-dimensional Urysohn width in higher dimensions.

Background

The paper places its volume-growth theorem within Gromov’s broader codimension-two program for positive scalar curvature. In addition to volume growth, this program predicts bounds for Urysohn width and macroscopic dimension. The cited conjecture asks whether a uniform positive scalar-curvature lower bound yields a corresponding bound on the (n−2)-dimensional Urysohn width. The paper notes that stronger estimates are known in dimension three, while the higher-dimensional form remains unresolved.