Does there exist a finite set of radii such that some aperiodic packing of the plane by disks of those radii is believed to achieve the maximal packing density (not achieved by any periodic packing)?

I would also be interested in a nonconstructive proof that such a set of radii must exist.

[I wanted to add “aperiodic” or “quasicrystal” or something like that as a tag; I settled for “almost-periodic-function”, but please re-tag as appropriate.]