  We consider the conformal flow model derived by Bizo\'n, Craps, Evnin, Hunik,
Luyten, and Maliborski [Commun. Math. Phys. 353 (2017) 1179-1199] as a normal
form for the conformally invariant cubic wave equation on $\mathbb{S}^3$. We
prove that the energy attains a global constrained maximum at a family of
particular stationary solutions which we call the ground state family. Using
this fact and spectral properties of the linearized flow (which are interesting
on their own due to a supersymmetric structure) we prove nonlinear orbital
stability of the ground state family. The main difficulty in the proof is due
to the degeneracy of the ground state family as a constrained maximizer of the
energy.
