Twisted Singer conjecture for closed aspherical odd-dimensional manifolds

Let $M$ be a closed aspherical $(2n+1)$-manifold. For a cohomology class $\phi\in H^1(M;\mathbb{R})$, let $b_i^{(2)}(\widetilde M;\phi)$ denote the twisted $L^2$-Betti numbers of the universal cover, and let $\chi^{(2)}(\widetilde M;\phi)$ denote the corresponding twisted $L^2$-Euler characteristic. Twisted Singer conjecture. One has

$$b_i^{(2)}(\widetilde M; {-}) \equiv 0$$

for all $i\ne n$. Moreover,

$$\chi^{(2)}(\widetilde M; {-})=(-1)^n\cdot b_n^{(2)}(\widetilde M; {-})$$

is a seminorm up to sign. This is a twisted analogue of the Singer conjecture, motivated by the vanishing result for virtually fibered cohomology classes; the conjecture concerns vanishing outside the lower middle dimension and the relationship between the remaining twisted $L^2$-Betti number and the Thurston norm.