Granville–Pappalardi conjecture for prime values with power bases

Let $a,b\geqslant 2$ be integers, and choose maximal positive integers $k$ and $\ell$ such that $a=A^k$ and $b=B^\ell$. Granville–Pappalardi's power-base conjecture. There are infinitely many primes of the form

$$|a^m-b^n|$$

with $m,n$ positive integers, unless there exists a non-zero integer $Q$ such that

$$(a^m-b^n,Q)>1$$

for all positive integers $m,n$ satisfying $(m,\ell)=(n,k)=1$; in that case, there are only finitely many primes of the form $|a^m-b^n|$. This extends the proposed prime-value statement to the case where one or both bases are perfect powers by restricting to exponents not covered by the forced factorization; the remaining assertion is conjectural.