Convergence to a stable equilibrium for random initial opinions

Consider the continuous-time opinion-dynamics system with $n$ agents, and choose the agents' initial opinions independently and randomly according to a bounded probability density function with connected support that is bounded below by a positive number on its support. Random convergence conjecture. The probability that the system converges to a stable equilibrium tends to $1$ as the number of agents tends to infinity.

The conjecture concerns the observed convergence behavior of the model: the paper proves convergence to clusters and characterizes stable equilibria, but does not establish that random initial conditions converge to a stable equilibrium with probability tending to one.