Does having unique normal forms imply weak normalization and confluence?
It is a little bit sketchy, but here is my argument: Suppose that there are three terms t, u, v such that t reduces to u, t reduces to v and suppose that each term has a unique normal form. Since you are weakly normalizing as you said, there exists a normal form for v and u. But since the normal form is unique, then v and u are joinable. Hence, the system is confluent.