The Strong Atiyah Conjecture over a subfield of the complex numbers

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$, let $G$ be a countable group with a bound on the orders of its finite subgroups, and let $\mathrm{lcm}(G)$ denote the least common multiple of the orders of the finite subgroups of $G$. For a finite matrix $A$ over $\mathbb{K}[G]$, let $\mathrm{rk}_{\mathcal N(G)}(A)$ be its von Neumann rank. The Strong Atiyah Conjecture over $\mathbb{K}$. For every finite matrix $A$ over $\mathbb{K}[G]$, one has

$$\mathrm{rk}_{\mathcal N(G)}(A)\in\frac{1}{\mathrm{lcm}(G)}\mathbb{Z}.$$

This conjecture is known over $\mathbb{C}$ for many classes of groups, including locally indicable groups, braid groups, elementary amenable groups, virtually compact special groups, and $3$-manifold groups, and it is stable under free products of groups satisfying it. It has applications such as implying Kaplansky's zero-divisor conjecture for torsion-free groups, but it remains open in general.