The conjectured convergence-rate bound for the disagreement component

Let $\vec{z}_t$ denote the disagreement component of the opinion vector at time $t$, and let $\lambda$ be the parameter appearing in the model. Here $\mathbb{E}$ denotes expectation, $\|\cdot\|$ the relevant norm, and $\lesssim$ an inequality up to a constant independent of $t$. Convergence-rate conjecture. Simulations suggest that

$$\mathbb{E}[\|\vec{z}_t\|^2] \lesssim \begin{cases} \frac{1}{t^{2 \lambda}} & \lambda \leq \frac{1}{2} \\ \frac{1}{t} & \lambda > \frac{1}{2} \end{cases}.$$

This would quantify the rate at which the disagreement component vanishes and identify a change in the predicted rate at $\lambda=\frac12$. The paper presents it as a direction for future work based on simulations; no proof or resolution is supplied.