$n\ge3$ is 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
$$\sum_{j=1}^{n}\left|\sum_{I\subseteq \{1,2,3,\cdots,n\},|I|=j}\prod_{k\in I}z_{k}\right|^2\ge 1$$