Finitely generated non-nilpotent unimodularly closed groups

Determine whether there exists a finitely generated non-nilpotent unimodularly closed group, where a group is unimodularly closed if every unimodular equation over it has a solution within the group.

Background

The paper defines a group to be unimodularly closed when every unimodular equation over the group has a solution in the group itself. It proves that finitely generated solvable or linear groups are unimodularly closed exactly when they are nilpotent, and lists connected compact Lie groups and Hall’s universal locally finite group as further examples of unimodularly closed groups. It then explicitly asks whether a finitely generated example can be non-nilpotent.