How small can the nonzero sum of $O(\log n)$ distinct $n-$th roots of unity be?

Question

The OEIS sequence oeis.org/A108380 gives the least number of distinct n-th roots of unity summing to the smallest possible nonzero magnitude.

This sequence seems to imply that the least number of distinct $n-$th roots of unity summing to the smallest possible nonzero magnitude is growing with something like linear growth with $n.$

I have asked about Fourier coefficients of characteristic functions of sets with a geometric series structure, that is sets of the form $\{1,2,\ldots,2^{s-1}\}$ with $n\geq 2^{s-1}$ implying $s=O(\log n),$ see the link below.

Answering the question in the title would provide a lower bound to the linked question when$n$ is a prime.

Lower bound on geometric series structured sets' Fourier coefficients

Related more general question:

How small can a sum of a few roots of unity be?

A pigeonholing argument shows it can be as small as exponential in $-(\log{n})^2$. It should be bounded below at least by $e^{-o(n)}$ as $n \to \infty$, but already this is an unsolved problem, even for sums of five $n$-th roots of unity. Non-trivial upper bound can be proved for the number of extremely small sums. — Vesselin Dimitrov
– Vesselin Dimitrov, Commented Jan 9, 2017 at 18:27
There is a lower bound by norms: There are at most $n$ conjugates of this sum, each of size at most $\log n$, and the product of the conjugates is a nonzero integer, hence at least $1$, so the number is at least $1/(\log n)^n = e^(-n \log \log n)$. Of course this is quite far from the upper bound and conjectural lower bound that Vesselin Dimitrov gave. — Will Sawin
– Will Sawin, Commented Jan 9, 2017 at 18:52

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How small can the nonzero sum of $O(\log n)$ distinct $n-$th roots of unity be?

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