Covering-radius conjecture for the odd trigonometric moment curve

Let $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the odd trigonometric moment curve embedding, let $\Pi_{2k+1}:\mathbb{S}^{2k+1}\to\gamma_{2k+1}(\mathbb{S}^1)$ be the nearest-point projection, and let $d_{2k+1}$ be the spherical distance. Define the covering radius by

$$\rho(\Pi_{2k+1})=\max_{q\in\mathbb{S}^{2k+1}}\min_{t\in\mathbb{S}^1}d_{2k+1}(q,\gamma_{2k+1}(t)).$$

Let $\delta_k$ be the lower-bound quantity used earlier in the paper. Covering-radius conjecture.

$$\rho(\Pi_{2k+1})\leq\frac{\delta_k}{2}=\frac{\pi k}{2k+1}.$$

The estimate is proposed as part of the strategy for proving optimality of the TMC-EPC; the supplied text gives no proof and presents the problem as open.