The empirical limit measure dimension conjecture for branching random walks on hyperbolic groups

Let $\Gamma$ be a non-elementary hyperbolic group generated by a finite set $S$, and let $\nu$ be a probability measure on $S$ with $\operatorname{supp}(\nu)=S$. Consider the branching random walk on $\Gamma$ with growth rate $\beta>0$ induced by the random walk with transition probabilities $p(x,y)=\nu(x^{-1}y)$. Let $v>0$ be the almost-sure escape rate, let $a>1$ be the visual parameter of the boundary metric $d_a$ on $\partial \Gamma$, and let $\mu_\infty$ be the empirical limit measure on the accumulation set of the branching random walk. Empirical limit measure dimension conjecture. In this setting, almost surely,

$$\dim \operatorname{supp} \mu_\infty=\left(\frac{\beta}{v\log(a)}\right)\wedge \dim \partial \Gamma.$$

This predicts that the dimension of the support of the empirical limit measure is determined by the ratio of the branching growth rate to the random walk's escape rate and the visual scaling parameter, capped by the dimension of the boundary. The claim is proposed for branching random walks on non-elementary hyperbolic groups; the source does not state a resolution.