$l$-th power radioprimal conjecture

Question

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 $\varepsilon>0$.

Answer 1

Assuming the strengthened version of Hardy-Littlewood conjecture I discuss here (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 $k_i(n)=a_i$.