A conditional result on Goldbach numbers in short intervals

Assume the Riemann Hypothesis and a weaker form of Montgomery's pair correlation conjecture, i.e., for every $\theta\in[1,2)$ $$F(X,T)=4\sum_{0<\gamma_1,\gamma_2\leq T}\frac{X^{i(\gamma_1-\gamma_2)}}{4+(\gamma_1-\gamma_2)^2} \ll T(\log T)^\theta,$$ where $\gamma_j$, $j=1,2$, run over the imaginary part of the non-trivial zeros of the Riemann zeta-function, holds uniformly for $\frac{X}{H}\leq T\leq X$, where $1\leq H \leq X$. Then, for all sufficiently large $X$ and $H\gg (\log X)^\theta$, we have that the interval $[X,X+H]$ contains a even integer which is a sum of two primes