Topological conjugacy in the moving-head model

Determine the computational difficulty of deciding topological conjugacy for elements of the length-preserving local-rewrite group G^u_{X,S} acting in the moving-head model.

Background

The paper studies topological conjugacy for elements of G^u_X and related groups in the moving-tape action, obtaining undecidability results through a reduction from the halting problem. It distinguishes this from the moving-head model, in which a one-head machine acts on configurations that explicitly include the head position. The authors state that they have not investigated the topological conjugacy problem in that model.

The open problem asks for the computational complexity of deciding whether two elements of G^u_{X,S} are topologically conjugate under the moving-head action. The paper only conjecturally indicates that the problem is at least undecidable and does not establish any formal lower or upper complexity classification.