Optimality conjecture for trigonometric moment curve correspondences

Let $k\in\mathbb{N}$, and let $\gamma_{2k}:\mathbb{S}^1\to\mathbb{S}^{2k}$ and $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the trigonometric moment curve embeddings defined in the paper. Let $R_{2k}$ and $R_{2k+1}$ be their associated TMC-EPC correspondences, and let $\mathrm{dis}$ denote correspondence distortion. TMC optimality conjecture.

$$\mathrm{dis}(R_{2k+1})=\mathrm{dis}(R_{2k})=\frac{2\pi k}{2k+1}$$

for all $k\in\mathbb{N}$. The preceding lower bound shows that these distortions are at least the displayed value, while computational experiments for $k=1,2,3,4$ suggest optimality; the general assertion remains open.