Existence and uniqueness conjecture for Linnell division rings

Let $K$ be a field and let $G$ be a torsion-free group. A Linnell division $K[G]$-ring is an epic division $K[G]$-ring that is $(N,G)$-free for every subgroup $N$ of $G$. Linnell division-ring conjecture. A Linnell division $K[G]$-ring exists and is unique up to $K[G]$-isomorphism. This is proposed as a variation of the strong Atiyah conjecture for torsion-free groups, and the cited passage gives no resolution.