Stability classification of collinear relative equilibria in positive curvature

Let $\mu\in(0,\mu_{\operatorname{T}})$ be the mass ratio, let $\kappa>0$ be the curvature parameter, and use center-saddle, elliptic, complex-saddle, and Lyapunov stable in their standard linear or dynamical meanings for relative equilibria.

Collinear stability classification conjecture. $\operatorname{L}_1,\operatorname{L}_2,\operatorname{E}_3$, and $\operatorname{A}_1$ are center-saddles whenever they exist; $\operatorname{E}_2$ is Lyapunov stable and $\widetilde{\operatorname{L}}_3$ is elliptic whenever they exist; and $\operatorname{L}_3$ is a center-saddle for $0<\kappa<\kappa_4(\mu)$, undergoes a subcritical pitchfork bifurcation at $\kappa=\kappa_4(\mu)$, and is elliptic for all larger $\kappa$ for which it exists.

The source presents this as conjectural because rigorous stability proofs are unavailable in portions of the parameter space and the change of stability of $\operatorname{L}_3$ has not been rigorously tied to $\kappa_4(\mu)$.