Assuming Goldbach's conjecture, let's denote by $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ . Then Cramer's model allows to write that the probability $ P((n-k,n+k)\in\mathbb{P}^{2})=\dfrac{c_{k}}{\log^{2}n} $ and $ r_{0}(n)=\inf\{m,\sum_{k=0}^{m}\dfrac{c_{k}}{\log^{2}n}\geq 1\} $.
Would a proof that $ \forall k, c_{k}>0 $ entail that $ r_{0}(n)=O(\log^{2}n) $?