Connelly's uniformity conjecture for ternary disk packings

A disk packing is a packing of disks in the Euclidean plane, and its uniformity is the ratio of the smallest disk size to the largest disk size. A triangulated packing is one whose contact structure is triangulated; the hexagonal compact packing (HCP) is excluded from the comparison. The ternary triangulated packing numbered $53$ in the cited classification and its deformation are the configurations described in the source.

Connelly's uniformity conjecture. The ternary triangulated packing numbered $53$ is the most uniform triangulated packing other than HCP. Its proposed deformation yields the highest possible uniformity among all disk packings other than HCP.

The conjecture seeks the extremal size ratio among non-HCP packings, both within triangulated packings and without that restriction. The source reports numerical records for packing $53$ and for its deformation, but does not state that either extremal claim has been proved or disproved.