Kaplanski's zero divisor conjecture for group rings

Let $G$ be a torsion-free group and let $R$ be an integral domain. A non-trivial zero divisor in the group ring $R[G]$ is a nonzero element $\alpha\in R[G]$ for which there exists a nonzero $\beta\in R[G]$ such that $\alpha\beta=0$. Kaplanski's zero divisor conjecture. The group ring $R[G]$ does not contain non-trivial zero divisors. Progress concerning the conjecture has been made in various directions, including restrictions on possible zero divisors over the rationals, but the conjecture itself remains open.