If a term rewriting system is confluent, then every term has at most one normal form. Is the converse also true, or is confluence a strictly stronger property? I.e. if every term has at most one normal form, does the rewriting system need to be confluent? My intuition is that the answer is no; for example, it seems to me that you could theoretically have a term rewriting system in which every non-variable term has NO normal form (and thus, at most one normal form), but which is NOT confluent, because you might have one term that has infinite reductions in two different 'directions'. But this is just intuition, not a formal proof or counterexample.
Any help would be appreciated!