This is a (rephrased) repost of a question I asked on MSE about 6 months ago, but didn't receive a definitive answer.

Let $S(n)$ be the sum of the digits of $n$ (in base $10$), is it true that for every $n$ which is not a power of $10$ there are infinitely many $k$ such that $S(n^k)>S(n^{k+1})$? (here $n$ and $k$ are clearly both integers)

The case $n=2$ is answered in this MSE question, what can be said about other values of $n$?