Monotonicity conjecture for distortion of odd trigonometric moment curves

For each positive integer $k$, let $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the odd trigonometric moment curve embedding, and let $\mathrm{dis}(\gamma_{2k+1})$ denote its distortion. TMC distortion monotonicity conjecture. The function

$$k\longmapsto\mathrm{dis}(\gamma_{2k+1})$$

is monotonically increasing with $k$. Numerical experiments support this assertion for the tested values, while the paper notes that the distortion appears never to exceed approximately $1.77$; no general proof is given.