Karabegov’s conjecture on finite planar point configurations with three closest neighbors

Let a finite planar point configuration be a finite set of points in the plane, where the closest neighbors of a point are the points minimizing Euclidean distance from it. Karabegov’s conjecture. Any finite planar point configuration in which every point has exactly $3$ closest neighbors must contain at least $16$ points. The conjecture concerns the minimum size of a finite planar configuration whose nearest-neighbor graph is 3-regular; the source exhibits a configuration with $16$ points, while the asserted lower bound is not established in the supplied text.