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We study the stability of the critical points of the Onsager energyfunctional with Maier-Saupe interaction potential in general dimensions. Weshow that the stable critical points must be axisymmetric, which solves aproblem proposed by [H. Wang, P. Hoffman, Commun. Math. Sci. 6 (2008),949--974] and further conjectured by [P. Degond, A. Frouvelle, J.-G. Liu,Kinetic and Related Models, 15(2022), 417--465]. The main ingredients of theproof include a suitable decomposition of the second variation around thecritical points and a detailed analysis of the relation between the intensityof interaction and the order parameter characterizing the anisotropy of thesolution.

On Stability of Critical Points for the High Dimensional Onsager Functional with Maier-Saupe Potential