Almost perfect inhomogeneous powers in arithmetic progression

Let $S$ be a finite set of primes and write ${\mathbb{Z}}_S$ for the set of those non-zero integers whose prime divisors belong to $S$. Hajdu proved that the abc conjecture implies that the number of terms of any arithmetic progression in $H_S=\{\eta x^l\mid \eta \in {\mathbb{Z}}_S,x,l\in \mathbb{Z},\text{ with }x>0\text{ and }l\ge 2\}$ is bounded. Moreover, if $k\ge 3$ and the exponents of the powers are all $\ge 4$, then the number of such progressions are finite. We consider other sets and prove similar statements for these sets.