Naber’s k-symmetry conjecture for tangent cones of noncollapsed Ricci limit spaces

Prove that, in any noncollapsed n-dimensional Ricci limit space (X,d), for each point x and integer k, all tangent cones at x are k-symmetric away from a set of Hausdorff dimension strictly less than k−1.

Background

This conjecture would provide quantitative stratification and symmetry control for tangent cones in noncollapsed Ricci limit spaces. The authors explain that, combined with their Theorem 1.4(ii), it would imply the Colding–Naber Conjecture 1.2 on topological stability of tangent cone cross-sections.

Establishing the k-symmetry bound would refine singular set estimates and rigidity phenomena across dimensions.