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 Gaĺois radius $r$ of type $(a,b)$ has a Galois radius $r'$ of type $(b,a)$.