A conjecture of Erdös on 3-powerful numbers

Erdös conjectured that the Diophantine equation $x+y=z$ has infinitely many solutions in pairwise coprime 3-powerful integers, i.e., positive integers $n$ for which $p\mid n$ implies $p^3\mid n$. This was recently proved by Nitaj who, however, was unable to verify the further conjecture that this could be done infinitely often with integers $x$, $y$ and $z$ none of which is a perfect cube. This is now demonstrated.