Robust-regime conjecture for the splitting model

Let $\eta_n^h$ be the initial configuration of the splitting model with parameter $h$ and size parameter $n$, and let $\mathbf B_{c r}$ denote the ball appearing in Theorem

$$, whose conclusion asserts the corresponding ball-type behavior of the model. **Robust-regime conjecture.** Theorem$$

holds for all

$$\nh<\frac{1}{2}.$$

The paper proves the theorem for $h<0$ and conjectures its extension to the full range $h<1/2$. The authors explain that the standard abelian-sandpile argument does not directly apply because the splitting model is non-abelian.