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 $\varepsilon>0$.



$l$-th power radioprimal conjecture
Assuming the strengthened version of Hardy-Littlewood conjecture I discuss [here](https://mathoverflow.net/a/395878/30186) (which follows from Dickson's conjecture), the following much stronger result holds: let $a\_0,\dots,a\_m$ be any sequence of natural numbers with $a\_0\geq 1$ and $a\_{i+1}\geq a\_i+2$ for $i<m$. Then there are infinitely many integers $n$ such that $k\_i(n)=a\_i$ for $i\leq m$.


Let $N$ be some fixed natural number to be chosen later. Consider the set $T=\{N,2N,\dots,(a\_m-m)N\}\cup\{-(a\_i-i)N\mid 0\leq i\leq m\}$, and $S$ the set of other numbers in the interval $[-(a\_m-m)N,(a\_m-m)N]$. If $N$ is divisible by all small enough primes, then $T$ is an admissible tuple. Let $n$ be any integer given by the strengthened Hardy-Littlewood conjecture above for these $T,S$. Then clearly its first $m+1$ primality radii are $(a\_i-i)N$ for $0\leq i\leq m$, and you can count there are $a\_i$ primes between $n-(a\_i-i)N$ and $n+(a\_i-i)N$, so $k\_i(n)=a\_i$.


