The exact asymptotic formula conjecture for Turán's problem 10

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

$$F_n(\alpha)=\inf_{z_k\in\mathbb C,\,|z_k|\geq1}\max_{\nu=1,\ldots,\lfloor\alpha n^2\rfloor}\left|\sum_{k=1}^n z_k^\nu\right|.$$

Theorem $u77$ gives

$$(\underline A(\alpha)-o(1))\sqrt n\leq F_n(\alpha)\leq(\overline A(\alpha)+o(1))\sqrt n,$$

where

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

Exact asymptotic formula conjecture. One can choose $A(\alpha)=\underline A(\alpha)=\overline A(\alpha)$ in Theorem $u77$. This would determine the true asymptotic constant for every $\alpha>0$. The theorem establishes matching bounds only in special cases, notably $\alpha=1$; the general problem remains open.