Subpolynomial relative asymptotic dimension of real hyperbolic space

Let $d$ be the dimension of real hyperbolic space $\mathbb{H}^d$. Let $o(\operatorname{poly}(d))$ denote the family of non-decreasing functions $f$ such that

$$\lim_{t\to\infty}\frac{f(t)}{t^d}=0.$$

Subpolynomial asymptotic-dimension conjecture.

$$\operatorname{asdim}_{o(\operatorname{poly}(d))}(\mathbb{H}^d)=d.$$

The paper describes this as an optimality statement for a preceding theorem. A weaker equality with $\operatorname{poly}(d-1)$ would already have consequences for excluding certain regular maps from $\mathbb{H}^d$ to products involving trees, but the conjectured subpolynomial statement itself is left unresolved in the supplied text.