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.

Added on 6/4/20: I just discovered https://arxiv.org/pdf/1111.4917.pdf ("Densest binary sphere packings" by Hopkins, Stillinger, and Torquato) which asserts "In $R^2$, periodic, quasicrystalline, and directionally periodic structures can all be found among the putative densest binary disk packings [38, 50–52]." However I have not yet looked at the references for details. I'll update this post again if I learn more.

[Note to administrators: 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.]