On the asymptotic formula for Goldbach numbers in short intervals

Let $R(k)=\sum\limits_{l+m=k}\Lambda(l)\Lambda(m)$, $\Sing(k) = 2 \prod\limits_{p>2}\left(1-\frac{1}{(p-1)^2}\right) \prod\limits_{\substack{ p\mid k\\ p>2 }} \left(\frac{p-1}{p-2}\right)$ if $k$ is even and $\Sing(k) =0$ if $k$ is odd. It is known that $R(k) \sim k\Sing(k)$ as $N\to \infty$ for almost all $k\in [N,2N]$ and that $$ \sum_{k\in [n,n+H)}R(k) \sim \sum_{k\in [n,n+H)} k\Sing(k) \quad\hbox{for} \quad n\to \infty \eqno{(1)}$$ uniformly for $H\geq n^{1/6+\epsilon}$. Here we prove, assuming $N^\epsilon\leq H\leq N^{1/6+\epsilon}$ and $N\to\infty$, that (1) holds for almost all $n\in [N,2N]$.