Group rings whose unitary units are nilpotent

Let F be a field of characteristic different from 2 and G a group. Under the classical involution on the group ring FG, we show that if FG is modular, then the group of unitary units of FG is nilpotent if and only if the entire unit group is nilpotent. We also demonstrate that this does not necessarily hold if FG is not modular, but it is still true if F is algebraically closed.