Equal-mass conjecture for central configurations of homothetic polygons

Let a central configuration consist of $L$ homothetic regular polygons, with the masses assigned to the vertices of each polygon. Equal-mass conjecture. Such a central configuration is possible only if the masses in each polygon are equal. The preceding results establish this in cases with $L\leq 2$ and in the general case when the polygon radii are sufficiently different; the conjecture concerns the remaining configurations.