The mean-field conjecture for critical exponents of quasi-isometric trees

Let $\Gamma$ be an infinite $d$-regular tree, and let $\Gamma'$ be a graph quasi-isometric to $\Gamma$, including a decoration of $\Gamma$. A critical exponent is the power-law exponent governing avalanche-mass probabilities. Mean-field critical-exponent conjecture. The graph $\Gamma'$ has the same critical exponent as $\Gamma$, namely $\frac{3}{2}$. This proposes that quasi-isometric decorations preserve the mean-field exponent known for infinite regular trees. The paper presents it as a conjecture and does not establish it in general.