Mean-square comparison conjecture for prime progressions

Let $E(x;q,a)$ be the discrepancy

$$E(x;q,a)=\left|\pi_{q,a}(x)-\frac{\pi(x)}{\varphi(q)}\right|,$$

where $\pi_{q,a}(x)$ counts primes at most $x$ congruent to $a$ modulo the prime $q$. Mean-square comparison conjecture. There exists an $\eta\in[0,1)$ such that, for prime $q$,

$$E(x;q,1)^2\ll q^\eta\cdot\frac1q\sum_{\substack{a=1\\(a,q)=1}}^m E(x;q,a)^2.$$

This alternative conjectural control of the progression error term is presented as a possible hypothesis for extending the paper's agreement between number-theoretic calculations and the Ratios Conjecture prediction to arbitrary finite support.