# Is the conjunction of Goldbach and NFPR conjecture actually equivalent to Hardy-Littlewood k-tuple conjecture?

In this previous question of mine I introduce under Goldbach's conjecture the notation $r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ as well as the related so-called NFPR conjecture which states that $\forall\varepsilon>0,\forall x>2,\max \{r_{0}(n),n\leq x\}\ll_{\varepsilon}x^{\varepsilon}$, from which I derive an asymptotics for $\lim\inf_{n\to\infty} p_{n+k}-p_{n}$ that is a consequence of Hardy-Littlewood k-tuple conjecture.

I would like to know whether the conjunction of Goldbach and NFPR conjectures is actually equivalent to Hardy-Littlewood's or if a more precise, $\varepsilon$ free upper bound for $r_{0}(n)$ is needed : for example do we need a small enough $\alpha$ such that $r_{0}(n)<C\log^{\alpha}n$ for all sufficiently large $n$ and some absolute positive constant $C$ and if so would $\alpha=2$ do the job ?