Spectral instability conjecture for periodic homographic motions of planar strong-force systems

Consider the planar $N$-body problem with masses $m_k>0$ under a $1/r^\alpha$ potential, where $0<\alpha<2$, and let $q$ be a central configuration. A periodic homographic motion is a solution of the form $z(t)q$; the two classes considered are those near the circular relative equilibrium solution and those near the homothetic “total collapse” solution. Spectral instability conjecture. At least one of these two classes of periodic homographic motions is spectrally unstable. This concerns the relationship between sectional curvature and linear stability in the planar $N$-body problem; the supplied text does not state whether the claim has been proved or disproved.