Stability conjecture for random initial opinions in the Hegselmann–Krause model

Let the initial opinions $x_1(0),\dots,x_N(0)$ be chosen independently and uniformly from $[0,L]$, and let $q_{L,N}$ be the probability that the resulting equilibrium is stable, meaning that every pair of distinct positive-weight clusters is separated by more than one. Stability conjecture. For every fixed $L$, $q_{L,N}\to 1$ as $N\to\infty$. This asserts that random initial opinions almost surely produce stable equilibria in the large-population limit; the source further notes a stronger expected version for any continuous bounded density with connected support, while attributing a related result to prior work.