The large sieve for square moduli under Hooley's hypothesis $R^*$

Let $S(Q,M,N,(a_n)):=\sum_{q\le Q}\sum_{(a,q)=1}|\sum_{M<n\le M+N}a_ne(an/q^2)|^2$ be Zhao's large sieve sum with square moduli. At the critical point $N=Q^3$ the best known unconditional bound, due to Baier and Zhao (2008), is $S\ll Q^{1/2+\varepsilon}N\sum |a_n|^2$, against the conjectured $Q^{\va