Conjugacy of reversible cellular automata and one-head machines

We show that conjugacy of reversible cellular automata is undecidable, whether the conjugacy is to be performed by another reversible cellular automaton or by a general homeomorphism. This gives rise to a new family of finitely-generated groups with undecidable conjugacy problems, whose descriptions arguably do not involve any type of computation. For many automorphism groups of subshifts, as well as the group of asynchronous transducers and the homeomorphism group of the Cantor set, our result implies the existence of two elements such that every finitely-generated subgroup containing both has undecidable conjugacy problem. We say that conjugacy in these groups is eventually locally undecidable. We also prove that the Brin-Thompson group $2V$ and groups of reversible Turing machines have undecidable conjugacy problems, and show that the word problems of the automorphism group and the topological full group of every full shift are eventually locally co-NP-complete.