Gottschalk's surjunctivity conjecture for cellular automata

Let $G$ be a group and $A$ a finite alphabet. A cellular automaton is a $G$-equivariant continuous map $T\colon A^G\to A^G$. Gottschalk's conjecture. If $T$ is injective, then it is surjective and hence an isomorphism. This conjecture motivates the notion of a surjunctive group: a group for which the assertion holds for every finite alphabet; it is known for all sofic groups, while the general case remains open.