A strong form of Goldbach's conjecture

Let $G(n)$) denote the number of pairs of primes whose sum is the even integer $n$. For even integers $n\geq 6$, the proposed lower bound is

$$\begin{aligned} G(n) &> \sum_{k=3}^{n/2} \frac{1}{\ln(k)\,\ln(n-k)}\\ &\quad - \left|\mathcal{O}\left(\sum_{k=3}^{n/2} \frac{1}{[\ln(k)]\,[\ln(n-k)]^2}\right)\right|\\ &\quad - \left|\mathcal{O}\left(\sum_{k=3}^{n/2} \frac{1}{[\ln(k)]^2\,[\ln(n-k)]}\right)\right|. \end{aligned}$$

A strong form of Goldbach's conjecture. For every even integer $n\geq 6$,

$$G(n)>\sum_{k=3}^{n/2} \frac{1}{\ln(k)\,\ln(n-k)}-\left|\mathcal{O}\left(\sum_{k=3}^{n/2} \frac{1}{[\ln(k)]\,[\ln(n-k)]^2}\right)\right|-\left|\mathcal{O}\left(\sum_{k=3}^{n/2} \frac{1}{[\ln(k)]^2\,[\ln(n-k)]}\right)\right|>0.$$

This would give a quantitative lower bound for the Goldbach partition function and, in particular, imply that every even integer at least $6$ is a sum of two primes. The paper presents it as a heuristic conjecture derived from a probabilistic use of the prime number theorem; no resolution is supplied.