Optimal distortion conjecture for the TMC projection

Let $\psi_{2k+1}:\mathbb{S}^{2k+1}\to\mathbb{S}^1$ and $\psi_{2k}:\mathbb{S}^{2k}\to\mathbb{S}^1$ be the projections associated with the trigonometric moment curve embeddings, and let $\mathrm{disc}$ denote projection distortion. TMC projection distortion conjecture.

$$\mathrm{disc}(\psi_{2k+1})=\mathrm{disc}(\psi_{2k})=\delta_k.$$

The paper gives a lower bound of $\delta_k$ for both quantities, motivating the conjectured equality; no resolution is stated.