Has the following conjecture been prooved, or has any step in the direction of its proof been done?

"ANY Probabilistic Cellular Automata converge fast on the stationary probability distribution iff the infinite system is ergodic and converge slowly iff the infinite system is non-ergodic".

"Fast" means that the distance between the asymptotic probability distribution and the probability distribution at time $t$ decreases exponentially with time. "Slow" if the speed of convergence is slower than exponential.