Uniform Markov-constant bound for square-torus disk packings

Let $N\geq 6$, let $\delta$ be the density of a packing of $N$ equal disks in a square torus, and define its Markov constant by

$$M=\frac{\pi}{2}\frac{1}{\left(\frac{\pi}{\sqrt{12}}-\delta\right)N}.$$

Square-torus Markov-bound conjecture. Every such packing satisfies $M\leq 6.25$. Equivalently,

$$\frac{\pi}{2\sqrt{3}}-\delta>\frac{\pi}{2}\left(\frac{1}{6.25N}\right).$$

The conjecture would give a uniform upper bound on the Markov constant and hence a uniform lower bound on the density gap.