The saturation conjecture for densest packings of discs with finitely many radii

Consider packings by $n$ different given discs. A packing is saturated if no further disc can be added. Saturation conjecture. If the densest compact packing is saturated, then the maximal density among all packings is attained by a compact packing. This extends the paper's result for binary disc packings to any number of disc radii; the saturation hypothesis is needed starting with three disc radii, and the supplied text does not establish the conjecture in general.