We prove that any cone having vertex density strictly between 1 and 3, occurring as the weak limit of a sequence of oriented, immersed, stable minimal hypersurfaces and lying near a pair of hyperplanes must itself be a pair of hyperplanes.
"A Rigidity Theorem for Stable Minimal Hypercones." J. Differential Geom. 68 (3) 433 - 514, Nov 2004. https://doi.org/10.4310/jdg/1115669592