Stability conjecture for lattice systems with elastic collisions

Let $N+1$ particles form a lattice model with elastic collisions. Define

$$A_k:=\sum_{i=1}^k a_i,\qquad B_k:=\sum_{i=1}^k b_i,\qquad U_k:=\frac{B_k-A_k}{k},$$

so that $U_k$ is the speed of the centre of mass of the leftmost $k$ particles, while $U_{N+1}$ is the speed of the full system's centre of mass. Elastic-collision stability conjecture. The system consists of a single stable cloud if and only if

$$U_k>U_{N+1}\quad\text{for all }k\in[N].$$

Moreover, if the system is stable, it satisfies a strong law of large numbers with limiting speed $U_{N+1}$. This is motivated by the corresponding stability condition in the diffusion model, but the paper presents it as a conjecture for the lattice model.