The conjecture on the variance of primes in short intervals

Let $\psi(x)=\sum_{n\leqslant x}\Lambda(n)$ be the Chebyshev function. Short-interval variance conjecture. For all $\varepsilon>0$, if $h\leqslant X^{1-\varepsilon}$, then

$$\int_0^X\big(\psi(x+h)-\psi(x)-h\big)^2\,dx=(1+o_\varepsilon(1))hX\log(X/h)$$

as $X\to\infty$. This conjecture gives the expected variance of the number of primes in short intervals and, under the Riemann Hypothesis, is equivalent to Montgomery's pair correlation conjecture; it remains open.