Dihedral-family conjecture for non-axisymmetric critical points of the Onsager potential

Consider the Onsager potential with interaction parameter $\mu>0$. For a positive integer $n$, let $D^{2n\mathrm{h}}$ denote the relevant horizontal dihedral symmetry group, and call a critical point non-axisymmetric when it lacks axial symmetry. Dihedral-family conjecture. There exists a non-axisymmetric critical point with $D^{2n\mathrm{h}}$ symmetry for some $\mu>0$. Numerical computations exhibit several non-axisymmetric states with discrete symmetries, while the asserted existence for the stated dihedral family is not proved in the supplied material.