Convergence of the population's average opinion toward the leaders' average opinion

Let $f(t,A,w)$ be the population distribution, let $f_\ell(z)$ be the leaders' opinion distribution, and define the population's average opinion and the leaders' average opinion by

$$m_w(t)=\int_{\mathbb{R}\times I}w f(t,A,w)\,\mathrm{d}A\,\mathrm{d}w,\qquad \mu_\ell=\int_I z f_\ell(z)\,\mathrm{d}z.$$

Assume that the interaction function satisfies $G\equiv 1$. Convergence conjecture. The average opinion of the population $m_w(t)$ converges toward the average opinion of the leaders $\mu_\ell$ as $t\to+\infty$:

$$\lim_{t\to+\infty}m_w(t)=\mu_\ell.$$

The conjecture concerns the long-time influence of leaders on the population's mean opinion; the preceding evolution equation shows that, for $G\equiv1$, the mean is driven by its difference from the leaders' average opinion. No resolution is supplied in the source.