The Manna model relation between stationary, transition and threshold densities

Consider the Manna model on $\mathbb Z$. Let $\rho_s$ be its stationary density, let $\rho_{tr}$ be the transition density identified by the change in the long-time toppling behavior, and let $\rho_{th}$ be the threshold density for stabilization. Manna density-relation conjecture. The three densities satisfy

$$\rho_s=\rho_{tr}\neq\rho_{th}=1.$$

The conjecture is motivated by the observed quasi-stationary state of the fixed-energy Manna model and by numerical findings indicating that $\rho_{tr}$ is strictly below $1$, whereas stabilization eventually occurs at the threshold density. The relation differs from the behavior established for the other models discussed in the source.