Stability conjecture for high-Markov-constant square-torus packings

For each integer $N\geq 2$, let $\epsilon(N)$ be a positive quantity with

$$\epsilon(N)\longrightarrow 0$$

as $N$ tends to infinity. Let $M$ be the Markov constant of a packing of $N$ equal disks in a square torus, and interpret distance between packings using the configuration-space metric. High-Markov stability conjecture. For every $N=2,3,\dots$, there is such an $\epsilon(N)$ so that every packing with $M>3$ lies within $\epsilon(N)$ of a most dense packing. This predicts that high-Markov-constant packings become asymptotically close to optimal configurations.