Existence of aperiodic triangulated disk packings

A triangulated packing is a disk packing whose contact structure is triangulated, and the hexagonal compact packing is the standard equal-disk packing. A packing is periodic if it is invariant under a rank-two translation lattice. The disk sizes are drawn from a finite set.

Aperiodic disk-packing conjecture. There exists a finite set of disk sizes that admits a triangulated packing other than the hexagonal compact packing, while no such packing is periodic. In the weaker version, the densest such packing is required to be nonperiodic.

This is motivated by the relation between triangulated disk packings and tiling problems, and by the search for an aperiodic disk analogue of the Domino problem. The source presents this as a first step toward possible undecidability results and does not report a resolution.