The trivial idempotents conjecture for torsion-free group rings

Let $\Bbbk$ be a field of characteristic zero and let $G$ be a torsion-free group. An idempotent in the group ring $\Bbbk[G]$ is an element $p$ satisfying $p^2=p$. The trivial idempotents conjecture. Every idempotent in $\Bbbk[G]$ is trivial, meaning it is either $0$ or $1$. This conjecture is refuted: a counterexample would give a zero-divisor in $\Bbbk[G]$, and the source identifies this with a counterexample to Kaplansky's zero-divisor problem.