Critical Hausdorff dimension conjecture for branching random walks on hyperbolic groups

Let $\Gamma$ be a nonelementary hyperbolic group with word metric $d$, let $\partial\Gamma$ have a visual metric $d_a$ for some parameter $a>1$, and let $\mu$ be an admissible, superexponential, symmetric probability on $\Gamma$. Let $\rho$ be the spectral radius of the associated random walk, and let $\nu$ be an offspring distribution on $\mathbb{N}$ with mean $\lambda=\rho^{-1}$. Write $\Lambda$ for the limit set of ${\rm BRW}(\Gamma,\nu,\mu)$. Critical Hausdorff dimension conjecture. The limit set $(\Lambda,d_a)$ has Hausdorff dimension

$$\dim_H(\Lambda)=\log_a H(\rho^{-1})$$

almost surely. Below the critical point, the corresponding dimension formula is proved; at criticality, the lower bound is known and the conjecture asserts equality.