  We adapt the method of Simon [JDG '93] to prove a $C^{1,\alpha}$-regularity
theorem for minimal varifolds which resemble a cone $\bf{C}_0^2$ over an
equiangular geodesic net. For varifold classes admitting a "no-hole" condition
on the singular set, we additionally establish $C^{1,\alpha}$-regularity near
the cone $\bf{C}_0^2 \times \mathbb R^m$. Combined with work of Allard [Ann. of
Math. '72], Simon [JDG '93], Taylor [Ann. of Math. '76], and Naber-Valtorta
[Ann. of Math. '17], our result implies a $C^{1,\alpha}$-structure for the top
three strata of minimizing clusters and size-minimizing currents, and a
Lipschitz structure on the $(n-3)$-stratum.
