Simó–Yoccoz uniqueness conjecture for convex planar four-body central configurations

Let four bodies with positive masses be arranged in a specified ordering along the boundary of their convex hull. A planar configuration is convex if no body lies inside or on the convex hull of the other three bodies; otherwise it is concave. Simó–Yoccoz conjecture. There is a unique convex planar central configuration of the 4-body problem for each ordering of the masses in the boundary of its convex hull. MacMillan and Bartky proved existence of at least one such configuration for any four masses and any ordering, while uniqueness remains an open question; the conjecture has been attributed to Yoccoz and is also associated with Simó.