The 2R-Conjecture for cluster gaps in the Hegselmann–Krause model

Let $R>0$ be the interaction radius, and consider the Hegselmann–Krause dynamics in the restricted setting with $N$ points on a compact interval. A fixed point consists of clusters of co-located points, separated by gaps between consecutive clusters. The 2R-Conjecture. As $N$ tends to infinity, the gaps between the clusters in the fixed point are approximately $2R$. This is a classical qualitative conjecture, but the source notes that no precise meaning of “approximately” has yet been established in the literature, and the conjecture remains unresolved.