Finite-time constant-topology conjecture for SBC and SBI opinion dynamics

Let $n$ interacting agents have opinion vector $x(t)\in\mathbb{R}^n$ and bounds vector $r\in\mathbb{R}^n_{>0}$. In the synchronized bounded confidence (SBC) and synchronized bounded influence (SBI) systems, the state-dependent proximity digraph is denoted by $G_r(x(t))$. Constant-topology conjecture. For any trajectory $x(t)$ of an SBC or SBI system, there exists a finite time $\tau$ after which $G_r(x(t))$ remains constant. This would reduce the long-term dynamics to a fixed interconnection topology; the paper partly proves this conjecture, while universal convergence of all trajectories remains open.