Dimension bound for the nonuniqueness locus of tangent cones

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

Nonuniqueness-locus conjecture.

$$\dim_{Haus}(\mathcal{NU})\leq n-3.$$

This conjecture predicts that failure of uniqueness of tangent cones is confined to a set of codimension at least three in a noncollapsed Ricci-limit space. The examples preceding it show that tangent cones need not be unique, while the conjectured dimension estimate remains open.