The exact asymptotic formula conjecture for unit-circle Turán sums

Let $\alpha>0$ be constant, and define

$$G_n(\alpha)=\inf_{z_k\in\mathbb C,\,|z_k|=1}\max_{\nu=1,\ldots,\lfloor\alpha n^2\rfloor}\left|\sum_{k=1}^n z_k^\nu\right|.$$

Theorem $avs$ gives

$$(\underline B(\alpha)-o(1))\sqrt n\leq G_n(\alpha)\leq(\overline B(\alpha)+o(1))\sqrt n,$$

where

$$\underline B(\alpha)=\begin{cases}1,&0<\alpha\leq1,\sqrt{\frac32-\frac1{2\alpha}},&1\leq\alpha\leq3,\sqrt{2-\frac2\alpha},&3\leq\alpha,\end{cases} \qquad \overline B(\alpha)=\begin{cases}1,&0<\alpha\leq1,\sqrt2,&1<\alpha\leq2,\sqrt3,&2<\alpha\leq3,\\2,&3<\alpha.\end{cases}$$

Exact unit-circle asymptotic conjecture. One can choose $B(\alpha)=\underline B(\alpha)=\overline B(\alpha)$ in Theorem $avs$. This conjecture would close the gap between the known lower and upper asymptotic bounds for unit-modulus points. The bounds coincide for $0<\alpha\leq1$, while the general case remains open.