Best upper bound for $ r_{0}(n) $ under Goldbach and Chowla's conjectures

Question

Assume Goldbach's conjecture. Then for any large enough composite integer $ n $ $ r_{0}(n) : =\inf\{r\ge 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ exists and is obviously smaller than $ n $ . Does the assumption of Chowla's conjecture imply that for all $k>0$ one has $\sup_{n\le x}\{r_{0}(n)\}=O(x^{1/k})$?