Dimension bound for the locus of tangent cones with differing homeomorphism types

Let $Y^n$ be a noncollapsed limit of Riemannian manifolds with lower Ricci bounds. Let $\mathcal{NH}\subseteq Y$ be the set of points where the tangent cones at the given point are not of the same homeomorphism type.

Homeomorphism-type conjecture.

$$\dim_{Haus}(\mathcal{NH})\leq n-5.$$

This conjecture gives a stronger codimension estimate for points whose tangent cones differ in homeomorphism type, a phenomenon exhibited by the examples discussed in the paper. The dimension bound remains open.