The rates of growth in an acylindrically hyperbolic group

Let $G$ be an acylindrically hyperbolic group on a $delta$-hyperbolic space $X$. Assume there exists $M$ such that for any generating set $S$ of $G$, $S^M$ contains a hyperbolic element on $X$. Suppose that $G$ is equationally Noetherian. Then we show the set of the growth rate of $G$ is well-ordered. The conclusion is known for hyperbolic groups, and this is a generalization. Our result applies to all lattices in simple Lie groups of rank-1, and more generally, some family of relatively hyperbolic groups. A potential application is a mapping class group, to which the theorem applies if it is equationally Noetherian.