Given a set of distinct real numbers $(x_j)$ and non-zero complex $(c_j)$, then the large sieve says that

$$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\left|\sum_{j}c_j e^{2\pi i n x_j}\right|^2\leq \sum_{j}|c_j|^2.$$

Are there lower bounds saying that

$$\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\left|\sum_{j}c_j e^{2\pi i n x_j}\right|^2\geq \varepsilon \,\,?$$

I am aware of the paper titled Lower bounds for expressions of large sieve type, but this deals with a different formulation than the one I am interested in. Namely, it only refers to the dual equality and it includes more terms than I do. Any help would be greatly appreciated.