The Invertibles Conjecture for group rings

Let $K$ be a division ring and let $G$ be a group. A one-sided invertible element of $K[G]$ is an element having either a left or a right inverse. The Invertibles Conjecture. If $K[G]$ contains a one-sided invertible element that is not of the form $kg$ for $g\in G$ and $k\in K$, then $G$ has torsion. This conjecture is presented as a well-known conjecture about group rings and is attributed in the source to a statement of V02 for $K=\mathbb{C}$; its resolution is not specified in the supplied text.