Stable-domain conjecture for eternal Galton–Watson tree dimensions

Let $[\boldsymbol{T},\boldsymbol{o}]$ be a unimodular eternal Galton–Watson tree. Suppose its offspring distribution is in the domain of attraction of an $\alpha$-stable distribution, with $\alpha\in[1,2]$. Stable-domain conjecture. Then

$$\operatorname{udim}_M(\boldsymbol{T})=\operatorname{udim}_H(\boldsymbol{T})=\frac{\alpha}{\alpha-1}.$$

This extends the finite-variance case, where the Minkowski dimension is proved to equal $2$, and the source indicates that the corresponding Hausdorff-dimension result is proved later. The endpoint $\alpha=1$ makes the displayed expression undefined, so the formulation should be checked against the original paper.