The equality of unrestricted and unit-circle asymptotic constants conjecture

For $\alpha>0$, let $A(\alpha)$ be the asymptotic constant conjectured for

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

and let $B(\alpha)$ be the corresponding constant when $|z_k|=1$. Equality of asymptotic constants conjecture. One has

$$A(\alpha)=B(\alpha).$$

The equality would mean that allowing points outside the unit circle does not improve the asymptotic constant. The unit-circle restriction is important in the available Fejér-kernel method, and extending the result to $|z_k|\geq1$ is described as difficult; the conjecture remains open.