The mean-field conjecture for cell-wise first-wave critical exponents

Let $\Gamma$ be an infinite $d$-regular tree, and let $\Gamma'$ be a graph quasi-isometric to $\Gamma$, including a decoration of $\Gamma$. The cell-wise first-wave critical exponent is the power-law exponent for the number of cells that topple in the first wave of an avalanche. Cell-wise first-wave mean-field conjecture. The cell-wise first-wave critical exponent of $\Gamma'$ equals the first-wave critical exponent of $\Gamma$, namely $\frac{3}{2}$. This is a proposed analogue of the critical-exponent conjecture that is more tractable for decorated trees; the paper proves the value for the expanded cactus but leaves the general quasi-isometric case open.