I would like to know if some widely believed conjecture, be it GRH, Hardy-Littlewood conjecture, or any other would imply the following statement for some $l>1$:

$l$-th power radioprimal growth conjecture ($l$-PRG conjecture for short)

Let $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $r_{i+1}(n):=\inf\{r>r_{i}(n),(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{i}(n):=\pi(n+r_{i}(n))-\pi(n-r_{i}(n))$. There exists $C_{l}>1$ such that there are infinitely many $n$ such that $i\leq C_{l}$ implies $k_{i-1}(n)=i^{l}$.

I would also like to know if $C_{l}$ can be taken arbitrarily large, and if the hypothetical existence of a maximal $l$ such that the $l$-PRG conjecture holds implies $r_{0}(n)<K(\varepsilon)\log^{l+\varepsilon} n$ for all sufficiently large $n$ and all $\varepsilon>0$.