Existence classification of collinear relative equilibria in positive curvature

Let $\boldsymbol{p}_1$ and $\boldsymbol{p}_2$ be the primaries of the curved restricted three-body problem, let $\boldsymbol{L}_i$, $\boldsymbol{E}_i$, $\boldsymbol{A}_1$, and their tilded counterparts denote the named collinear relative equilibria, and let $\boldsymbol{\mu}_{\operatorname{T}}$ be the stated threshold mass ratio. For a fixed $\mu\in(0,\mu_{\operatorname{T}})$, the curvature parameter is $\kappa>0$.

Existence classification conjecture. There exist $\kappa_i=\kappa_i(\mu)\in(0,\pi^2/4)$ with $\kappa_1<\kappa_2<\kappa_3$ such that the curved restricted three-body problem has exactly the following collinear relative equilibria: six, $\operatorname{L}_1,\operatorname{L}_2,\operatorname{L}_3,\operatorname{E}_2,\operatorname{E}_3,\operatorname{A}_1$, for $\kappa<\kappa_1$; four, $\operatorname{L}_1,\operatorname{L}_3,\operatorname{E}_3,\operatorname{A}_1$, for $\kappa_1<\kappa<\kappa_2$; two, $\operatorname{L}_1,\operatorname{A}_1$, for $\kappa_2<\kappa<\kappa_3$; and four, $\operatorname{L}_1,\widetilde{\operatorname{L}}_3,\widetilde{\operatorname{E}}_3,\operatorname{A}_1$, for $\kappa_3<\kappa<\pi^2/4$. The values $\kappa_i$ correspond respectively to the coalescence and disappearance of $\operatorname{L}_2,\operatorname{E}_2$ and of $\operatorname{L}_3,\operatorname{E}_3$, and to the emergence of $\widetilde{\operatorname{L}}_3,\widetilde{\operatorname{E}}_3$. Moreover, the distances of $\operatorname{E}_2,\operatorname{E}_3,\operatorname{A}_1$ from the rotation center tend to infinity as $\kappa\to0^+$, while $\operatorname{L}_1,\operatorname{L}_2,\operatorname{L}_3$ converge to the planar Lagrange points.

The claim summarizes rigorous results, numerical observations, and computer-assisted evidence, but the source explicitly notes that complete existence proofs for all parameter values are still missing.