The Weak Atiyah Conjecture over a subfield of the complex numbers

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$, and let $G$ be a countable group with a bound on the orders of its finite subgroups. For a finite matrix $A$ over $\mathbb{K}[G]$, let $\mathrm{rk}_{\mathcal N(G)}(A)$ be its von Neumann rank. The Weak Atiyah Conjecture over $\mathbb{K}$. There exists an integer $n\in\mathbb{Z}$ such that, for every finite matrix $A$ over $\mathbb{K}[G]$,

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

This is a weakening of the Strong Atiyah Conjecture that retains the discreteness of the possible von Neumann ranks. The source describes it as wide open in general, although it is known for the groups established in the paper's theorem.