Finiteness of singularities for higher-weight axially symmetric minimizers of the relaxed Dirichlet energy

Let k≥2 be an integer. We prove that every minimizer of the Bethuel--Brezis--Coron relaxed Dirichlet energy in the signed meridional k-axial class, with continuous H¹ degree-zero data on the boundary of a ball, is smooth outside a finite subset of the symmetry axis. This extends the weight-one finiteness theorem of Hardt--Lin--Poon and resolves the finite-singularity question posed by Martinazzi--Rivière as Open Problem~A, in the fixed-boundary Bethuel--Brezis--Coron setting with continuous degree-zero data. The interior argument uses a distributional graph-boundary identity, height-preserving half-turn replacements, and exact square completions retaining the current cost. Cap-valued crosscuts and a finite incidence count exclude singular accumulation at the boundary poles.