Kaplansky's unit conjecture for torsion-free group algebras

Let $\mathbb{F}$ be a field and $G$ a torsion-free group. A unit of the group algebra $\mathbb{F}[G]$ is called trivial when it is a non-zero scalar multiple of a group element. Kaplansky's unit conjecture. The algebra $\mathbb{F}[G]$ contains no non-trivial units; equivalently, every unit is a non-zero scalar multiple of an element of $G$. This is a central open problem in the theory of group algebras. The paper proves lower bounds on the support of an inverse when one factor has support of size $4$, but does not resolve the conjecture in general.