Does there exist a semigroup theory for which there is a reduced canonical term-rewriting system that is not length decreasing?

The source page states the original problem together with its recorded qualifications and progress updates as follows.

Does there exist a semigroup theory (without constants in the equations) for which there is a reduced canonical term-rewriting system (with the right-hand side and subterms of the left in normal form) that is not length decreasing?