Conjecture on first excited states in the general-mass N-body problem

Consider the N-body problem with general positive masses and its relative equilibria, namely solutions generated by central configurations. The first excited state is the relative equilibrium with the relevant lowest excited energy. Conjecture on first excited states. The first excited states are the co-linear relative equilibria for general masses. The source proves the analogous statement for the equal-mass three-body problem and recalls Moulton's theorem on co-linear central configurations, but does not establish the asserted general-mass conclusion.