  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 closely follows that giving the
boundary regularity result of Hardt and Simon.
