Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$.

Does it suffice to prove that the number $N_{Gal}(n)$ of Galois radii of $n$ is greater than $f(n):=\sum_{p\leq\sqrt{n}}\lfloor\frac{\log n}{\log p}\rfloor$ to deduce $n$ has at least a primality radius? If I'm not mistaken one has $f(n)=O(\sqrt{n})$.