# $l$-th power radioprimal conjecture

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 $$r_{0}(n) for all sufficiently large $$n$$ and all $$\varepsilon>0$$.

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 $$i. Then there are infinitely many integers $$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$$.