Atiyah's conjecture for torsion-free groups

Let $G$ be a torsion-free group, let $A$ be a finite $n\times m$ matrix over $\mathbb ZG$, and consider the right-multiplication map

$$\bigoplus_n \ell^2(G)\longrightarrow\bigoplus_m \ell^2(G),\qquad x\longmapsto xA.$$

Atiyah conjecture. The von Neumann dimension of the kernel of this map is an integer.

The conjecture concerns the possible von Neumann dimensions arising from matrices over the integral group ring and would imply strong integrality for $L^2$-homological calculations. It is open; the source records cases proved for extensions of free groups by elementary amenable groups and for residually torsion-free elementary amenable groups, and notes stability under subgroups and free products.