Solving this problem would bridge computable structure theory, harmonic analysis, and model theory in a fundamental way. A positive answer for broad classes of groups would give logicians new tools to transfer decidability results between algebraic and analytic settings, while a negative answer or undecidability result would sharply delineate the limits of algorithmic reasoning about continuous mathematical structures. It would also inform the study of cohesive powers and Tennenbaum-style phenomena for analytic structures, clarifying when computable or cohesive approximations to such algebras can exist.