The rationality and integrality conjecture for $L^2$-Betti numbers

Let $\Gamma$ be a group and let $X$ be a free finite $\Gamma$-CW-complex. For each degree $p$, let $b_p^{(2)}(X;\mathcal{N}(\Gamma))$ denote the corresponding $L^2$-Betti number. Let $d$ be a positive integer such that the order of every finite subgroup of $\Gamma$ divides $d$.

Rationality and integrality conjecture. For every degree $p$,

$$b_p^{(2)}(X;\mathcal{N}(\Gamma))\in\mathbb{Q}.$$

Moreover,

$$d\,b_p^{(2)}(X;\mathcal{N}(\Gamma))\in\mathbb{Z}.$$

The assertion concerns the possible arithmetic values of $L^2$-Betti numbers for finite free $\Gamma$-complexes. The source gives no evidence of resolution status.