Pseudo-stable behavior conjecture for SBC and SBI systems

Let $x(t)\in\mathbb{R}^n$ be a trajectory of an SBC or SBI system, and let a trajectory have pseudo-stable behavior after a time $\tau$ when the agents split into non-empty fixed and converging subsets, with fixed agents already at their limiting opinions and each converging agent moving monotonically toward its limiting opinion. Pseudo-stable behavior conjecture. For any SBC or SBI trajectory, there exists a finite time after which the trajectory either reaches a fixed state or exhibits pseudo-stable behavior. The paper states that this conjecture is proved assuming the finite-time constant-topology conjecture.