Bounded Engel and solvable unitary units in group rings

Let FG be the group ring of a group G over a field F. We consider the group of unitary units of FG with respect to the classical involution. Under suitable restrictions upon F, we show that if the unitary units of FG are both bounded Engel and solvable, then the entire unit group of FG is nilpotent. This extends a result of Fisher, Parmenter and Sehgal.