Triangulation conjecture for continued-fraction square-torus packings

Let $N$ equal disks be packed in a square torus. A packing is said to correspond to a convergent of $1/\sqrt{3}$ or $2-\sqrt{3}$ when it is one of the Type I or Type II packings, respectively, associated with that continued-fraction convergent. Let the packing graph have the disk centers as vertices and packing contacts as edges. Triangulation conjecture. If the packing corresponds to one of these convergents, then its packing graph is a subgraph of a triangulation of the torus in which every vertex has degree six. This predicts a locally hexagonal combinatorial structure for the continued-fraction constructions.