Say $r$ is a Galois radius of level $l=ab$ and of type $(a,b)$ of $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime. Let $r_{l,0}(n)$ the smallest non negative Galois radius of $n$ of level $l$ and $r_{(a,b),0}(n)$ the smallest non negative Galois radius of $n$ of type $(a,b)$.
Fix some $d\in [\min(a,b),\max(a,b)]$ and denote by $$K_{d,0}(n):=\lim_{\varepsilon\to 0}\frac{1}{2}\left(\sharp\{s^k\in (n-r_{l,0}(n)-\varepsilon,n+r_{l,0}(n)+\varepsilon)\mid s\in\mathbb{P},k\leq d\}+\sharp\{s^k\in (n-r_{l,0}(n)+\varepsilon,n+r_{l,0}(n)-\varepsilon)\mid s\in\mathbb{P}, k\leq d\}\right)$$
Can one prove the inequality $\frac{2r_{l,0}(n)}{K_{d,0}(n)}\lesssim\log^{a+b}n$ for some values of $a$, $b$ and $d$?
Note that the special case $l=1$ is Cramer's conjecture.
