Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ is an inversion shift of $r$ if either $r-s$ or $r+s$ is a Galois radius of $n$ of type $(b,a)$. Let $s_{0}(n,r)$ be the smallest positive inversion shift of $r$.

Does any unbalanced Galois radius of any large enough integer have an inversion shift? Does one have $s_{0}(n,r)\ll_{\varepsilon}r^{1+\varepsilon}$?

More precisely, is the following conjecture true?

Conjecture: for every couple of different positive integers $(a,b)$, there is an integer $N_{a,b}$ such that every integer $n$ greater than $N_{a,b}$ which has a Galois radius $r$ of type $(a,b)$ has a Galois radius $r'$ of type $(b,a)$.

Edit March 9th 2023: we may consider the special case where $a$ and $b$ have the same Hamming weight $H$ (as the number of $1$'s in the binary representation of an integer) and write them as a block of $0$'s and $1$'s of given length $L$. The set $\mathcal{P}_{L,H}$ of permutations of such integers forms a finite group $G$. Among its elements, those of order $2$ may correspond bijectively to the set of inversion shifts of Galois radii of type $(a,b)$ of a given set of integers $n$.