Classification conjecture for positive-mass planar equilateral 5-body central configurations

Consider planar equilateral 5-body central configurations with all masses positive in the Newtonian case. The regular pentagon and the star are the configurations with equal masses, and the paper has also identified the symmetric configurations discussed immediately beforehand. Classification conjecture. The only positive-mass planar equilateral 5-body central configurations in the Newtonian case are the regular pentagon and star with equal masses together with those symmetric configurations. The statement is presented as a conjecture, and the surrounding discussion restricts the possible configurations to several geometric regions; a complete proof of the classification is not supplied.