Density classification conjecture for the GKL, Kari traffic, and majority-traffic rules

Let a cellular automaton or probabilistic cellular automaton evolve configurations in the binary state space on the one-dimensional lattice $\mathbb{Z}$. The GKL cellular automaton and Kari traffic cellular automaton are the rules described above, and the majority-traffic probabilistic cellular automaton has parameter $\alpha$. A model classifies the density if, for an independent initial configuration with density $p$, its trajectories converge weakly to $1^{\mathbb{Z}}$ for $p>1/2$ and to $0^{\mathbb{Z}}$ for $p<1/2$. Density classification conjecture. The GKL cellular automaton, the Kari traffic cellular automaton, and the majority-traffic probabilistic cellular automaton with

$$0<\alpha<\alpha_c$$

for some $0<\alpha_c\leq 1/2$, classify the density. These rules are among the best-performing candidates for density classification on finite rings, but the asserted classification on the infinite lattice remains unproved in the source.