Friedlander–Granville conjecture on primes in arithmetic progressions

Let $x\geq 1$, let $q\leq x$, and let $a$ be a residue class modulo $q$. Define the prime-counting error term $\mathcal{E}(x;q,a)$ by

$$\sum_{\substack{n \leq x \\ n \equiv a \pmod q}} \Lambda(n)=\frac{x}{\varphi(q)}+\mathcal{E}(x;q,a).$$

Friedlander–Granville conjecture. For every fixed $\varepsilon>0$,

$$\mathcal{E}(x;q,a)\ll \left(\frac{x}{q}\right)^{1/2}x^{\varepsilon}.$$

This conjecture predicts square-root-size errors uniformly for $q\leq x$ and is stronger than what the Generalized Riemann Hypothesis gives when $q$ is a positive power of $x$. Under the Generalized Riemann Hypothesis, Turán's result for the corresponding variance is consistent with this conjectural pointwise bound.