Let $(z_r)$ be a sequence of complex numbers and let $(\alpha_r)$ be real numbers with the property that $\|\alpha_r - \alpha_s\| \geq \delta > 0$ if $r \neq s$, where for $k \in \mathbb{R}$, $\|k\|$ denotes the distance from $k$ to the nearest integer.
Montgomery and Vaughan proved (see for instance https://en.wikipedia.org/wiki/Hilbert%27s_inequality) that $$ \left|\sum_{\substack{r,s \\ r \neq s}} \dfrac{z_r\overline{z_s}}{\sin \pi (\alpha_r - \alpha_s)}\right| \leq \delta^{-1} \sum_{r}|z_r|^2. $$
Question: Supposing we replace the $z_r\overline{z_s}$ above with a more general $f(s,r)\overline{f(r,s)}$ with $f : \mathbb{Z}\times \mathbb{Z} \to \mathbb{C}$, can we prove (or is it known) or disprove that $$ \left|\sum_{\substack{r,s \\ r \neq s}} \dfrac{f(s,r)\overline{f(r,s)}}{\sin \pi (\alpha_r - \alpha_s)}\right| \leq \delta^{-1} \sum_{r}|f(r,r)|^2. $$ Thanks!