Weak conjugacy conjecture for stable cellular automata limit sets

A stable cellular automaton is a cellular automaton whose sequence of iterated images stabilizes, and its limit set is the resulting invariant subshift. Two subshifts are weakly conjugate when there are factor maps from each subshift onto the other. A shift of finite type (SFT) is a subshift defined by finitely many forbidden finite patterns.

Weak conjugacy conjecture. The limit set of any stable cellular automaton is weakly conjugate to an SFT.

Characterizing the limit sets of stable cellular automata is an important open problem. Known constructions proceed through weak conjugacy with an SFT, and the conjecture would show that this accounts for every stable cellular automaton limit set.