Goldston–Montgomery conjecture on the variance of primes in short intervals

Let $\Lambda(n)$ denote the von Mangoldt function. For $\delta\in(0,1)$, set $H=X^\delta$. Goldston–Montgomery conjecture. As $X\to\infty$,

$$\frac{1}{X}\int_0^X\left(\sum_{x<n\leq x+H}\Lambda(n)-H\right)^2\,dx\sim H(\log X-\log H).$$

This predicts the variance of sums of the von Mangoldt function in short intervals and is connected with the local spacing of zeros of the Riemann zeta function. It is stronger than the Riemann hypothesis according to the source, but its resolution is not supplied here.