Relative asymptotic dimension of rank-one symmetric spaces

Let $X$ be a rank one symmetric space of noncompact type, and let $d$ be a parameter in the polynomial growth class. The notation $\overline{\operatorname{asdim}}_{\operatorname{poly}(d)}(X)$ denotes the upper relative asymptotic dimension used in the paper, and $\operatorname{asdim}(X)$ is the ordinary asymptotic dimension.

Rank-one symmetric-space conjecture. For every such $X$ and every $d$,

$$\overline{\operatorname{asdim}}_{\operatorname{poly}(d)}(X)=\operatorname{asdim}(X).$$

The authors note that this would in particular imply the analogous equality they expect for real hyperbolic spaces. The source does not report a proof or disproof.