  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.
