Under Goldbach's conjecture, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$ and $n_{r}:=\inf\{n>2,r_{0}(n)=r\}$.
Let $e_{0}(n):=\left(\frac{\log (2n)}{\log (2r_{0}(n))}\right)^{\frac{r_{0}(n)}{n}}$.
Does the following hold?:
$$\left(\dfrac{\pi(2n_{r})}{2n_{r}}\right)^{e_{0}(n_{r})}\sim\dfrac{k_{0}(n_{r})}{2r}$$