Large sieve conjecture for square moduli

Let $S$ be the set of squares, and let $S_t(Q/t)$ be the subset defined earlier in the paper. For $M\in\mathbb{Z}$, $N\in\mathbb{N}$, and a complex sequence $(a_n)$, one has

$$\sum_{q\in S_t(Q/t)}\sum_{\substack{a=1\gcd(a,q)=1}}^q\left|\sum_{n=M+1}^{M+N}a_n e\left(\frac{a}{q}n\right)\right|^2 \ll Q^{\varepsilon}\left(\frac{Q}{t}\left|S_t(Q/t)\right|+N\right)\sum_{n=M+1}^{M+N}|a_n|^2,$$

where the implied constant depends on $\varepsilon$. Square-modulus large sieve conjecture. The displayed inequality holds for all such parameters and sequences.