Finite-time constant-topology conjecture for heterogeneous Hegselmann–Krause dynamics

Let $r=\{r_1,\dots,r_n\}\in\mathbb{R}^n_{>0}$ be the agents' confidence bounds and let $x(t)\in\mathbb{R}^n$ evolve according to the heterogeneous Hegselmann–Krause system

$$x(t+1)=A(x(t))x(t),$$

where $G_r(y)$ is the proximity digraph with out-neighbors $\mathcal{N}_i(y)=\{j\in\{1,\dots,n\}:|y_i-y_j|\leq r_i\}$ and $A(y)$ averages the opinions over each agent's out-neighbors. Constant-topology conjecture. Along every trajectory, there exists a finite time $\tau$ such that

$$G_r(x(t))=G_r(x(\tau))\qquad\text{for all }t\geq\tau.$$

If the interconnection topology becomes fixed, the resulting linear averaging dynamics can be analyzed using fixed-topology methods; the conjecture would in particular imply convergence of every heterogeneous HK trajectory to a steady state. The paper reports extensive numerical evidence, but states that a proof remains open.