The division-ring embedding conjecture for torsion-free group algebras

Let $G$ be a torsion-free group and let $\mathbb{K}$ be a field. The division-ring embedding conjecture. The group algebra $\mathbb{K}[G]$ embeds into a division ring.

This is a strengthening of Kaplansky's zero-divisor conjecture, which only asserts that $\mathbb{K}[G]$ has no zero divisors. The source states that this strengthening is wide open; it is motivated by the fact that the Strong Atiyah Conjecture implies the zero-divisor conjecture for torsion-free groups.