Weighted dual quadratic large sieve conjecture

Let $r\geq 1$ be fixed, let $\delta>0$, and suppose that $H\geq n^{1/2+\delta}$. Put $M=H/\sqrt n$. Let $\mathcal{D}_{2r}(H,M;\omega)$ denote the weighted dual moment defined by

$$\mathcal{D}_{2r}(H,M;\omega)=\sum_{h\sim H}\left|\sum_{m\asymp M}V_{\omega;h,m}e\left(\frac{h^2}{4m}\right)\right|^{2r}.$$

Weighted dual quadratic large sieve conjecture. For every $\varepsilon>0$,

$$\mathcal{D}_{2r}(H,M;\omega)\ll_{r,\varepsilon,\delta}HM^rn^\varepsilon\Phi_r(\omega),$$

uniformly for $1\leq\omega\leq n/10$, with the dependence on $n$ suppressed in the notation. In the ideal case one expects $\Phi_r(\omega)\ll n^\varepsilon$.

This estimate is the missing higher-moment input suggested by the dual expansion. Establishing the ideal factor would yield the expected square-root-cancellation behavior in the dual variable, but the source does not state that it is known.