Classification conjecture for rigid two-geodesic square-torus packings

Let $N\geq 6$ and consider a rigid packing of $N$ equal disks in a square torus. Assume that its packing graph consists of two linear geodesics, and let $M$ be its Markov constant. Two-geodesic classification conjecture. If $M>3$, then the packing is a Type I or Type II packing, and it is maximally dense. The conjecture would classify all rigid packings in this two-geodesic regime whose Markov constant exceeds $3$.