In relation to this question: Relation between volume entropy and Hausdorff dim of limit set?

Given a metric space $Z$ and a hyperbolic approximation $X := hyp_{r_0}(Z)$ (as defined for example here).

I noticed the following correspondence (for $x_0 \in X$):

$$ \lim_{r \to \infty} \frac{1}{r} \log_{1/r_0}(vol(B_r(x_0))) = \dim_H(Z).$$

Is this generally true? Does this follow from the answer in the linked question?