Two dimensional covering systems and possible prime producing $a^m-b^n$

We exhibit a new application of two dimensional covering systems, examples of integer pairs $a,b$ for which $a^m-b^n$ has a prime divisor from some given finite set of primes, for every pair of integers $m,n\ge 0$. This leads us to conjecture what are the only possible obstructions to $\mathrm{\mid }{a}^m-b^n\mathrm{\mid }$ taking on infinitely many distinct prime values.