Polynomial lower-bound conjecture for sums of roots of unity

For an integer $N\ge 1$, let $f(n+1,N)$ be the least positive value of

$$\left|1+\zeta_1+\cdots+\zeta_n\right|$$

where $\zeta_1^N=\cdots=\zeta_n^N=1$.

Polynomial lower-bound conjecture. For given $n\ge 1$ there exist constants $c(n)>0$ and $\lambda(n)>0$ such that

$$f(n+1,N)\ge c(n)N^{-\lambda(n)}$$

for all $N\ge 1$.

Polynomial lower bounds are known in some small cases, while the general case, in particular for $n\ge 4$, is presented as unknown. The conjecture is described as folklore and would follow from a positive answer to a question posed by Myerson.