In classical opinion dynamics, we define $N$ agents. Agent $i$ has an initial opinion sampled from a distribution with mean $\mu$. Let agent $i$'s initial opinion be $y_i(0)$ (a random variable of that distribution). Agents interact with one another and update their opinions. The dynamics of the opinion update is given by $$ y(t + 1) = A y(t)$$ where $A \in R^{n \times n}$ is a row stochastic matrix and $y(t)$ is a vector composed of the opinions of all agents. If the opinions of each of the individual agents converge to some value, then it is easy to see that $$\lim_{t \rightarrow \infty} y(t) = Vy(0)$$ for some constant $V \in R^{n \times n}$.
If my understanding is correct, this paper (https://arxiv.org/pdf/2204.13610.pdf) claims that this is also true for the Hegselmann-Krause (HK) model (pg 6 remark 2). The HK model has the following dynamics. For each agent $i$ $$ y_i(t + 1) = \frac{1}{|N_i(t)|} \sum_{j \in N_i(t)} y_j(t)$$ where $N_i(t) = \{j| \mid y_i(t) - y_j(t) \mid < \epsilon\}$
I've been thinking about it and the problem I keep hitting is that the matrix $V$ for the HK model would be a function of the initial opinions, and thus not a constant. This means that the analysis done in the paper wouldn't work for the HK model because now we don't know whether $$\mathbb E[\lim_{t \rightarrow \infty} y_{col}(t)] = \mu$$ where $y_{col}(t) = \frac{1}{N}(\sum_i y_i(t))$
I'm not sure what I'm missing, I tried searching on the net for an hour for the paper that explicitly mentions this but I couldn't find it (not saying that it doesn't exist, but I'm probably not able to find it, or correctly understand). Any help would be appreciated. Thanks!