Chenciner's regular polygon conjecture for equal-mass circular central configurations

Let $n$ bodies have equal masses and lie on a common circle, with their center of mass coinciding with the center of the circle. A central configuration is one in which the acceleration vector of each body is a common scalar multiple of its position vector with respect to the center of mass. Chenciner's conjecture. The regular $n$-gon with equal masses is the unique central configuration such that all the bodies lie on a circle and the center of mass coincides with the center of the circle. This conjecture asks for a classification of circular central configurations under the stated equal-mass and center-of-mass conditions; the paper proves a uniqueness theorem only for co-circular four-body configurations with each cyclic ordering, so the general $n$-body assertion remains open.