The Strong 2R-Conjecture for Hegselmann–Krause cluster locations

For each $\lambda>0$, let $\Xi_{\lambda}^{(0)}$ be a simple stationary renewal process on $\mathbb{R}$ satisfying the functional strong law of large numbers, and let $\Xi_{\lambda}^{(t)}$ denote the Hegselmann–Krause evolution, with $\Xi_{\lambda}^{(\infty)}:=\lim_{t\to\infty}\Xi_{\lambda}^{(t)}$. Let $X_{\lambda}(i)$ be the number of points in the $i$-th limiting cluster. Strong 2R-Conjecture. The joint locations of the clusters of $\lim_{\lambda\to\infty}\Xi_{\lambda}^{(\infty)}$ should be distributed as $\theta+2\mathbb{Z}$, where $\theta\sim\mathrm{Unif}(-1,1)$, and the cluster sizes should satisfy

$$\left(\frac{1}{\lambda}X_{\lambda}(i)\right)_{i\in\mathbb{Z}}\xrightarrow{a.s.}(\ldots,2,2,2,\ldots).$$

The source presents this as a stronger, simulation-supported refinement of the weak 2R result: the expected gap converges to $2$, whereas this conjecture predicts a stationary lattice of cluster locations and normalized cluster sizes converging jointly to $2$.