Hooley's conjecture on the variance of primes in arithmetic progressions

For a fixed modulus $c$, define

$$G(X,c)=\sum_{\substack{A\bmod c\\ \gcd(A,c)=1}}\left|\sum_{\substack{n\leq X\\ n=A\bmod c}}\Lambda(n)-\frac{X}{\phi(c)}\right|^2,$$

where $\Lambda(n)$ is the von Mangoldt function and $\phi(c)$ is Euler's totient function. Hooley's conjecture.

$$G(X,c)\sim X\log c.$$

Hooley did not specify the size of $c$ relative to $X$ for which this asymptotic should hold. The source describes this conjecture as open, while noting that related asymptotic formulae are known when the modulus is averaged over a long range.