The homothety conjecture for infinity-shaped extremal curves

Let $\gamma_\alpha$ be a closed extremal curve determined by a pendulum trajectory $\alpha(t)$, and let $K(\xi)$ denote the complete elliptic integral of the first kind. For a nonzero integer $r$, consider $\alpha$ satisfying

$$\sin\frac{\alpha(t)}{2}=\xi\operatorname{sn}\Big(\frac{2rK(\xi)t}{\pi}\Big|\xi\Big).$$

Homothety conjecture. All infinity-shaped extremal curves are homothetic to the curve $\gamma_\alpha$ for this choice of $\alpha$, where $\xi\approx.90890856$.

This conjecture identifies the nontrivial extremal shape numerically obtained from the pendulum equation and predicts that all infinity-shaped extremal curves differ only by scaling and parametrization. The source provides numerical evidence but no proof or resolution.