On the ternary Goldbach problem with primes in independent arithmetic progressions

We show that for every fixed $A>0$ and $\theta >0$ there is a $\vartheta =\vartheta (A,\theta )>0$ with the following property. Let $n$ be odd and sufficiently large, and let $Q_1=Q_2:=n^{\text{h}}(\log n{)}^{-\vartheta }$ and $Q_3:=(\log n{)}^{\theta }$. Then for all $q_3\le Q_3$, all reduced residues $a_3$ mod $q_3$, almost all $q_2\le Q_2$, all admissible residues $a_2$ mod $q_2$, almost all $q_1\le Q_1$ and all admissible residues $a_1$ mod $q_1$, there exists a representation $n=p_1+p_2+p_3$ with primes $p_{i}\equiv a_{i} (q_{i})$,$i=1,2,3$.