Equality of distributional category and LS-category under non-negative Ricci curvature

Determine whether the distributional category dcat(M) equals the Lusternik–Schnirelmann category cat(M) for all closed manifolds M with non-negative Ricci curvature.

Background

The authors compare two lower bounds: Bazzoni–Lupton–Oprea’s inequality cat(M) ≥ b1(M) + e0(\widetilde M) for closed manifolds with non-negative Ricci curvature, and their own bound dcat(M) ≥ b1(M) + cu(W) derived from the Cheeger–Gromoll splitting M' ≅ T^k × W. Since W ≃ \widetilde M in this setting, these bounds suggest a possible equality cat(M) = dcat(M).

Motivated by this proximity, they explicitly raise the unresolved question of whether the two categories actually coincide for such manifolds.