Monotonicity conjecture for non-axisymmetric states in the coupled dipolar/Maier–Saupe potential

Fix $\sigma_0$ and suppose that the coupled dipolar/Maier–Saupe potential has a non-axisymmetric critical point at parameter $(\sigma_0,\kappa_0)$. Monotonicity conjecture. If a non-axisymmetric critical point exists for $(\sigma_0,\kappa_0)$, then that also exists for every $(\sigma_0,\kappa)$ with $\kappa>\kappa_0$. This predicts that the existence region of a non-axisymmetric state is upward-closed in the $\kappa$ direction; the supplied material gives numerical evidence but no proof.