Critical-threshold conjecture for the splitting model

Let the splitting model on $\mathbb Z^d$ have parameter $h$, and let robust and explosive denote its two regimes.

Critical-threshold conjecture. There exists a critical value $h_c=h_c(d)$ such that, for all $h<h_c$, the model is robust, while for all $h\geq h_c$, the model is explosive. Moreover,

$$\nh_c(2)=\frac{2}{3}.$$

The conjecture proposes that the robust and explosive behaviors are separated by a single threshold, independent of the splitting order. The value $h_c(2)=2/3$ is suggested by simulations, while the behavior in the intermediate parameter range is not established by the paper.