We investigate local minimizers defined on a bounded domain in the plane for a singular constrained energy functional that applies to Ball and Majumdar's modification of the Ginzburg-Landau Q-tensor model for nematic liquid crystals. We prove regularity of local minimizers and that their range does not intersect the boundary of the constraining set. In the case of liquid crystals, this implies that minimizers of the constrained Landau-de Gennes Q-tensor energy including all elasticity terms and a Maier-Saupe bulk term have eigenvalues that are strictly between -1/3 and 2/3 at all points.