Let $a,b$ be two loxodromic isometries of a CAT(0) space. Assume that, for every $n \geq 1$, $a^nb$ is also loxodromic. Is it possible for the translation length of $a^nb$ to be bounded independently of $n$?

First, I thought as obvious that the translation length of $a^nb$ has to tend to $+ \infty$ as $n \to + \infty$, but I may have been misled by the CAT(-1) case (where this is clearly true). Now, I go back and forth between a possible counterexample and an easy argument I am missing...