The equation $x^py^q=z^r$ and groups that act freely on $\Lambda$-trees

Let $G$ be a group that acts freely on a $\Lambda$-tree, where $\Lambda$ is an ordered abelian group, and let $x, y, z$ be elements in $G$. We show that if $x^p y^q = z^r$ with integers $p$, $q$, $r \geq 4$, then $x$, $y$ and $z$ commute. As a result, the one-relator groups with $x^p y^q = z^r$ as relator, are examples of hyperbolic and CAT($-1$) groups which do not act freely on any $\Lambda$-tree.