Under Goldbach's conjecture, let's denote by $r_{0}(n)$ for any large enough composite integer the smallest positive integer $r$ such that both $n+r$ and $n-r$ are prime, and by $k_{0}(n)$ the quantity $\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$, where $\pi(x)$ is the number of primes not exceeding $x$.
Define the "primal volume" $V(n)$ as $V(n):=2r_{0}(n)k_{0}(n)$. Is there an absolute constant $C$ such that $V(n)<\log^{C}n$? Do the inequalities $k_{0}(n)>\log^{2-\gamma}n$ and $2r_{0}(n)<\log^{2+\gamma}n$ simultaneously hold true for infinitely many $n$, where $\gamma=0.577215664901...$ is the Euler-Mascheroni constant?
I also formulate the following conjecture: Quantitative NFPR conjecture: $\lim\sup_{n\to\infty}\frac{\log V(n)}{\log\log n}=4$ and offer 200€ to the first person who will publish a proof or refutation thereof in a peer-reviewed journal.
Edit August 11th 2021: say $n$ is $k$-central if $k_{0}(n)=k$ and denote by $V_{k}(n)$ for such an integer $n$ the quantity $2kr_{0}(n)$, and by $M_{k}:=\lim\sup_{n\to\infty,k_{0}(n)=k}\frac{\log V_{k}(n)}{\log\log n}$. Then $M_{k}$ is an increasing function of $k$, and it would be interesting to figure out the value of $M_{1}$ and whether the sequence $(M_{k})_{k>0}$ converges or not.
Edit August 18th 2021: we can use the prime number theorem to approximate the primal volume $V(n)$ by $I(n):=C_{n}\left(\int_{P(n)}^{Q(n)}Li(x)dx-\pi(P(n))(Q(n)-P(n))\right)$ where $P(n):=n-r_{0}(n)$, $Q(n)=n+r_{0}(n)$ and $C_{n}$ is a positive quantity depending only on $n$ and "close" to $1$. Numerically, it seems that $C_{n}\approx\frac{1}{1-1/r_{0}(n)}$.