Conjecture that minimizing hypercones are strictly area-minimizing

A minimizing hypercone is a codimension-one cone that is area-minimizing, equivalently mass-minimizing. The paper excludes trivial examples in low dimensions. Strict area-minimality conjecture. Except for trivial examples in low dimensions, all minimizing hypercones are strictly area-minimizing. This is one of two long-standing conjectures concerning minimizing hypercones, raised by Simon; the paper studies its relationship with the realization of tangent cones in compact homologically area-minimizing singular submanifolds, but does not establish the conjecture in general.