Partial boundary regularity for co-dimension one area-minimizing currents at immersed $C^{1,\alpha}$ tangential boundary points

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,\alpha}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; this partial regularity is such that we can conclude the tangent cone is unique. The proof follows closely the boundary regularity result given by Hardt and Simon in [9].