Atiyah's integrality conjecture for $L^2$-Betti numbers

Let $X$ be a finite connected CW-complex with fundamental group $\Gamma$. If $\Gamma$ is torsion-free, then its $j$-th $L^2$-Betti number $b_j^{(2)}(X)$ is an integer.

Atiyah's integrality conjecture.

$$b_j^{(2)}(X)\in\mathbb{Z}.$$

The conjecture is known for certain classes of torsion-free groups, but remains open in general. It motivates the question of when the analogous $p$-adic Betti numbers are integral.