Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface.

Question. (i) Are the tangent cones at the origin regular, that is $\operatorname{sing} \mathbf{C} = \{ 0 \}?$ (ii) If one cone in $\operatorname{VarTan}(\lvert M \rvert,0)$ is regular, is there a 'direct' way of showing that they must all be regular? (By 'direct' I would mean without going via uniqueness of tangent cones à la Lojasiewicz–Simon.)

I believe (i) used to be an open question, but I am curious about its current status or partial progress. The Schoen–Simon regularity theory implies that the cones are either regular or cylindrical of the form $\mathbf{C}^7 \times \mathbf{R}$.

Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface.

Question. (i) Are the tangent cones at the origin regular, that is $\operatorname{sing} \mathbf{C} = \{ 0 \}?$ (ii) If one cone in $\operatorname{VarTan}(\lvert M \rvert,0)$ is regular, is there a 'direct' way of showing that they must all be regular? (By 'direct' I would mean without going via uniqueness of tangent cones à la Lojasiewicz–Simon.)

I believe (i) used to be an open question, but I am curious about its current status or partial progress. The Schoen–Simon regularity theory implies that the cones are either regular or cylindrical of the form $\mathbf{C}^7 \times \mathbf{R}$.

Correction. Otis Chodosh pointed out in his answer that this is incorrect. The aforementioned regularity theory of Schoen–Simon shows that the singular set of every tangent cone $\mathbf{C} \in \operatorname{VarTan}(\lvert M \rvert,0)$ is one-dimensional, but it can consist of several rays, one for each singularity in the link of the cone. It's the tangent cones taken at these rays that are cylindrical, of the form described above.