Axisymmetric-minimizer conjecture for the Onsager potential

Let $\mu$ be the Onsager interaction parameter, and consider the local and global minimizers of the Onsager free-energy functional. A state is axisymmetric if its density is invariant under rotations about some axis; a prolate state is an axisymmetric critical point with two density peaks at two antipodal points. Axisymmetric-minimizer conjecture. Any local minimizer is axisymmetric. For $\mu>\mu_1$, the global minimizer is a prolate state—an axisymmetric critical point with two density peaks at two antipodal points. Numerical solution landscapes support the conjecture, but the supplied material does not establish it analytically.