Uniqueness conjecture for non-axisymmetric critical points of the coupled dipolar/Maier–Saupe potential

Let $(\sigma_0,\kappa_0)$ be a parameter pair for the coupled dipolar/Maier–Saupe potential, and identify critical points that differ only by a rotation. A critical point is non-axisymmetric if it is not invariant under an axial symmetry. Uniqueness conjecture. There exists at most one non-axisymmetric critical point (regardless of the rotation) for each parameter $(\sigma_0,\kappa_0)$, and this critical point is an index-2 critical point. The claim is based on numerical solution landscapes and remains unproved in the supplied material.