Assume Goldbach's conjecture. Then for every integer $n$ greater than one there exists a non negative integer $r$ such that both $n-r$ and $n+r$ are prime numbers. For a given $n$, let's denote $r_{0}(n)$ the smallest of these numbers $r$. Following the ideas developped in About Goldbach's conjecture, my question is: which non trivial upper bound can be otained for $r_{0}(n)$ through the Chinese remainder theorem?

Actually, I conjecture that $r_{0}(n)=O(\log^{2}n)$ (it's worth noticing that this would imply Cramer's conjecture) but several experts told me it was very unlikely that we would be able to establish such an upper bound in the near future. Nevertheless, maybe a weaker but still interesting bound can be obtained through a clever analysis. Thanks in advance for any insight.