Density-maximizing conjecture for selected unequal binary disk packings

Let $r_b\approx 0.217$ and $r_c\approx 0.369$ be the two disk-size ratios described in the paper, and let $r_1\approx 0.637$ be the ratio associated with the indicated triangulated binary packing. A binary packing uses disks of these two sizes; a periodic packing is invariant under a rank-two translation lattice; the hexagonal compact packing is the standard equal-disk hexagonal packing, and a relaxation is the deformation of the triangulated packing depicted in Fig. 6.

Density-maximizing conjecture. For the ratios $r_b$ and $r_c$, the periodic nontriangulated binary packing depicted in Fig. 7 maximizes density among all binary packings with the same size ratio. Moreover, for every ratio around $r_1$ for which the relaxation of the indicated triangulated binary packing is denser than the hexagonal compact packing—including every ratio in $[0.627,0.645]$—that relaxation maximizes density.

The claim concerns whether locally strong, explicitly constructed packings attain the global density maximum beyond the triangulated setting. The source reports that the relevant upper and lower density bounds are very close, but gives no resolution of the conjecture.