The famous "unit conjecture" for group rings states that all units of a group ring $K[G]$ are trivial for a field $K$ and a torison-free group $G$. We are far away from solving the conjecture (See e.g. http://people.maths.ox.ac.uk/craven/docs/seminars/220909slides.pdf).

I wonder whether more is known about unitary units: Given a group homomorphism $w\colon G \rightarrow \mathbb{Z}/2\mathbb{Z}=\left\{\pm 1\right\}$ we can define an involution on $K[G]$ via $g\in G \mapsto \overline{g}=w(g)g^{-1}$ . The unitary units are those elements $x$ of the group ring with $\overline{x}x=1$. Do we know they are trivial for torsion-free groups?