Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f_n\colon S^{n-1}\to\mathbb R$ by $$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}|$$ Denote $z^*_n := \min_{S^{n-1}} f_n$.

Is it true that for sufficiently large $n$s (which depend on $S$), we have $z_n^* = 2(n-1)$?

*Remarks*:

$f_n(x)$ is sum of the absolute values of eigenvalues of a circulant matrix generated by $(0,x_1,x_2,\ldots,x_{n-1})$.

$f_n(x) = 2(n-1)$, when $x$ is all ones vector.

Above question stems from this question and may be related to the Littlewood problem, as a comment of Tao to the last linked question.