Isolation conjecture for mean curvature flow singularities

Consider the outer or inner flow of an embedded surface through singularities. A singularity is an isolated singularity if it is isolated in the relevant spacetime singular set. Isolation conjecture. All singularities are isolated unless an entire tube shrinks to a closed curve. The conjecture is motivated by the principle that level set flow behaves in some respects like an analytic function. Weaker formulations ask for finitely many singular times or generic isolation; the conjecture remains open.