Positive-Hausdorff-dimension fixed-point density conjecture for binary tree groups

Let $T$ be the complete infinite rooted binary tree, and let $G\leq\operatorname{Aut}(T)$ be level-transitive if it acts transitively on every level. Define its Hausdorff dimension by

$$h(G)=\lim_{n\rightarrow\infty}\frac{\log_2(\#G_n)}{\log_2(\#\operatorname{Aut}(T_n))},$$

where $T_n$ is the complete binary rooted tree of height $n$ and $G_n$ is the image of $G$ in $\operatorname{Aut}(T_n)$. Let $\mathcal{F}(G)$ be the fixed-point density introduced in the surrounding discussion. Positive-Hausdorff-dimension fixed-point density conjecture. If $G$ is level-transitive and $h(G)>0$, then

$$\mathcal{F}(G)=0.$$

The conjecture would simplify the computation of the corresponding fixed-point densities for arboreal groups; no proof or disproof is given in the source.