Existence classification of triangular relative equilibria in positive curvature

Let $\mu\in(0,\mu_{\operatorname{T}})$ be the mass ratio and let $\kappa>0$ be the curvature parameter. Denote the classical triangular relative equilibria by $\operatorname{L}_4$ and $\operatorname{L}_5$, and the additional triangular equilibria by $\operatorname{T}_1$ and $\operatorname{T}_2$.

Triangular existence classification conjecture. There exist $\kappa_4(\mu)<\kappa_5(\mu)$ in $(0,\pi^2/4)$ such that there are two triangular relative equilibria, $\operatorname{L}_4$ and $\operatorname{L}_5$, for $0<\kappa<\kappa_4$; four triangular relative equilibria, two continuations of $\operatorname{L}_4,\operatorname{L}_5$ together with $\operatorname{T}_1,\operatorname{T}_2$, for $\kappa_4<\kappa<\kappa_5$; and no triangular relative equilibria for $\kappa_5<\kappa<\pi^2/4$. The equilibria $\operatorname{T}_1,\operatorname{T}_2$ are created when $\operatorname{L}_3$ undergoes a subcritical pitchfork bifurcation at $\kappa=\kappa_4$, while simultaneous saddle-node bifurcations at $\kappa=\kappa_5$ cause the pairs $(\operatorname{L}_4,\operatorname{T}_1)$ and $(\operatorname{L}_5,\operatorname{T}_2)$ to collide and disappear.

The classification is based on numerical observations and computer-assisted proofs, but the source states that rigorous existence proofs for the bifurcation curves remain incomplete for some parameter values.