A pair correlation hypothesis and the exceptional set in Goldbach's problem

Let $$E(X,H) = \vert \{2n\in[X,X+H] : 2n\ \hbox{is not a sum of two primes}\}\vert $$ be the exceptional set for Goldbach's problem in short intervals. We will assume the Generalized Riemann Hypothesis and that, for $(a,q)=1$, $\epsilon>0$ and $\theta\in(0,\frac12]$ fixed, $$ F(X,T;q,a)= \sum_{\chi_1,\chi_2 \pmod{q}}\hskip-0.5cm \chi_1(a)\overline{\chi}_2(a) \tau(\overline{\chi}_1)\tau(\chi_2) \sum_{\vert \gamma_1\vert ,\vert \gamma_2\vert \leq T}X^{i(\gamma_1-\gamma_2)} w(\gamma_1-\gamma_2) \ll q^2TX^\epsilon, $$ where $w(u)= \frac{4}{4+u^2}$, $\tau(\chi)$ denotes the Gauss sum and $\gamma_j$, $j=1,2$, run over the imaginary part of the non trivial zeros of $L(s,\chi_j)$, holds uniformly for $\frac{X^{1-\theta}}{q}\leq T \leq X$ and $q\leq X^\theta$. Under the previous hypotheses we prove, for every $\epsilon>0$ fixed, that $$E(X,X^{2\theta})\ll_\epsilon X^\epsilon,$$ i.e. for $\theta>\frac{\epsilon}{2}$ all even integers in any interval of the form $[X,X+X^{2\theta}]$ but $O(X^\epsilon)$ exceptions are a sum of two primes