Let $c_1, \ldots, c_k \in \mathbf N^+$ and $x_1,\ldots,x_k \in \mathbf Z \setminus \{0\}$. It is possible to prove by elementary means that $(\omega(c_1 x_1^n+\cdots+c_kx_k^n))_{n\ge 1}$ is a bounded sequence only if $|x_1|=\cdots=|x_k|$. (As usual, $\omega(x)$ is, for every non-zero $x \in \mathbf Z$, the number of distinct prime divisors of $x$, while $\omega(0) := \infty$.)

On the other hand, there seems to be more than a chance that the same conclusion may also come as an easy consequence of the Subspace Theorem (or any of its descendants). This sounds plausible to me, but I don't see how to proceed. So I thought to ask here, with the hope that the question is not completely trivial for the experts in the field and not totally uninteresting for the others.

Q. Can the statement made in the first paragraph of this post be obtained, in a more or less straightforward way, from the Subspace Theorem? If so, is a proof along these lines written down anywhere?