Atiyah conjecture for torsion-free countable groups

Let $G$ be a torsion-free countable group, and let ${\mathcal{D}}(\mathbb{C}G\subset {\mathcal{U}}G)$ denote the ring appearing in the Atiyah-conjecture construction. Atiyah conjecture. The ring

$${\mathcal{D}}(\mathbb{C}G\subset {\mathcal{U}}G)$$

is a skew field. This is the conjectural skew-field property underlying the paper's agrarian homology constructions; the supplied context does not state a resolution or the precise known cases, so the conjecture remains open here.