Continuity conjecture for the Hausdorff dimension of branching-random-walk limit sets

Consider a branching random walk (BRW) on a non-amenable Cayley graph. Let $P$ be the transition kernel of the underlying random walk, let $\rho(P)$ be its spectral radius, let $m$ be the mean offspring number, and let the limit set be the random subset of the geometric boundary consisting of ends visited infinitely often by the BRW.

Hausdorff-dimension continuity conjecture. The Hausdorff dimension of the limit set is continuous for $m\neq 1/\rho(P)$ and discontinuous at $m=1/\rho(P)$.

For $m>1/\rho(P)$ the BRW is recurrent and the limit set equals the full boundary. The source relates the conjecture to known results for homogeneous trees, but the stated general claim remains open.