Let $n\ge2$ be a given positive integer, and $z_{1},z_{2},\cdots,z_{n}\in \mathbb{C}$,such
$$|z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2\ge n.$$
Prove or disprove
$$f_{n}=\sum_{j=1}^{n}\left|\sum_{I\subseteq \{1,2,3,\cdots,n\},|I|=j}\prod_{k\in I}z_{k}\right|^2\ge 1$$

In the particular case when $n=2$, it can be proved that
\begin{align*}f_{2}&=|z_{1}+z_{2}|^2+|z_{1}z_{2}|^2=(|z_{1}|^2+|z_{2}|^2)+|z_{1}z_{2}|^2+2\Re(z_{1}\overline{z_{2}})\\
&\ge |z_{1}|^2+|z_{2}|^2+|z_{1}z_{2}|^2-2|z_{1}z_{2}|\\
&=(|z_{1}|^2+|z_{2}|^2-1)+(|z_{1}z_{2}|-1)^2\\
&\ge(|z_{1}|^2+|z_{2}|^2-1)\\
&\ge 1  
\end{align*}