Anderson–Cheeger–Colding–Tian manifold conjecture for three-dimensional Ricci limits

Let $(M,d)$ be a metric space arising as the Gromov–Hausdorff limit of a sequence of three-dimensional Riemannian manifolds satisfying conditions (a) and (b) from the paper.

Anderson–Cheeger–Colding–Tian conjecture. The metric space $(M,d)$ is itself a manifold.

In dimension two, the analogous assertion follows from a theorem of G. Perelman because a lower Ricci-curvature bound gives a lower sectional-curvature bound. In dimension four, examples show that such limits need not be manifolds; the three-dimensional case is the conjectural borderline discussed here.