Average prime-pair counting formula

Taking $r>0$, let ${\pi }_{2r}(x)$ denote the number of prime pairs $(p,p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that ${\pi }_{2r}(x)\sim 2C_{2r}{\mathrm{l}\mathrm{i}}_2(x)$ with an explicit constant $C_{2r}>0$. There seems to be no good conjecture for the remainders ${\omega }_{2r}(x)={\pi }_{2r}(x)-2C_{2r}{\mathrm{l}\mathrm{i}}_2(x)$ that corresponds to Riemann’s formula for $\pi (x)-\mathrm{l}\mathrm{i}(x)$. However, there is a heuristic approximate formula for averages of the remainders ${\omega }_{2r}(x)$ which is supported by numerical results.