Connelly's densest triangulated circle-packing conjecture

Let a circle packing of the triangulated plane be a packing whose tangency graph is a triangulation of the plane. Suppose that at least two circles have different sizes, and let $r$ be the ratio of the minimal radius to the maximal radius. The reference packing is the one shown in Figure 1 of the source, and its ratio is a root near $0.651$ of

$$89x^8+1344x^7+4008x^6-464x^5-2410x^4+176x^3+296x^2-96x+1.$$

Connelly's conjecture. There exists no circle packing of the triangulated plane with $r$ greater than the ratio attained by the reference packing shown in the figure.

This conjecture concerns the densest triangulated circle packing when radii lie in an interval and at least two radii are distinct. The source presents it as an open problem based on the known upper bound $q>0.743$ obstruction and the approximately $0.651$ packing constructed by Thomas Fernique.