The optimal packing conjecture for nine congruent circles on a square flat torus

Let ${\mathbb T}^2$ be the square flat torus, and consider arrangements of 9 points representing the centers of 9 congruent circles. Let $d$ denote the maximal possible minimum distance between two points in the torus.

Nine-point packing conjecture. There is one unique, up to isometry, optimal arrangement of 9 points in ${\mathbb T}^2$, shown in the paper's Figure 9, with

$$d=\dfrac{1}{\sqrt{5+2\sqrt{3}}}\approx0.3437.$$

This conjecture extends the paper's proved optimal-packing results for $N=6$, $7$, and $8$ and predicts both the exact optimal separation and uniqueness of the arrangement for $N=9$.