BiHölder smoothability of noncollapsed RCD(−2,3) spaces with Euclidean tangent cones

Prove that any noncollapsed RCD(−2,3) space whose tangent cones are Euclidean at every point is biHölder homeomorphic to a smooth Riemannian manifold.

Background

The authors' manifold recognition theorem yields a topological manifold structure under Euclidean tangent cone assumptions, but a biHölder equivalence to smooth structures is stronger and aligns with results known for Ricci limit spaces via Ricci flow in dimension three.

Confirming this conjecture would bridge synthetic and smooth settings and advance regularity theory for RCD spaces.

References

Conjecture 1.17. Any noncollapsed RCD(−2,3) space with Euclidean tangent cones is biHölder homeomorphic to a smooth Riemannian manifold.

— Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below (2405.03839 - Bruè et al., 6 May 2024) in Conjecture 1.17, Section 1.6 (Open questions)