Uniqueness conjecture for the toroidal penny graph embedding of $K_{3,3}$

A toroidal penny graph is a contact graph of congruent non-overlapping disks embedded on a flat torus, with vertices representing disk centers and edges representing tangent pairs. Consider the embedding of $K_{3,3}$ whose toroidal coordinates are given by

$$\left(\frac{13}{36},-\frac{13}{36}\right), \left(\frac{11}{36},\frac{1}{36}\right), \left(\frac{1}{36},\frac{11}{36}\right), \left(-\frac{13}{36},\frac{13}{36}\right), \left(-\frac{11}{36},-\frac{1}{36}\right), \left(-\frac{1}{36},-\frac{11}{36}\right).$$

Uniqueness conjecture for the $K_{3,3}$ embedding. The proposed embedding of $K_{3,3}$, with coordinates as described in the table, is unique up to an isometry.

The claim concerns the rigidity of the exhibited toroidal penny graph realization of $K_{3,3}$: no non-isometric embedding with the same graph is expected to exist. The source provides the embedding and its packing radius but gives no resolution of this uniqueness question.