Trace-range conjecture for unconditional twisted group-algebra completions

Let $\Gamma$ be a torsion-free group such that $(\Gamma,\sigma)$ satisfies the twisted Bost conjecture, and suppose that the classifying space $B\Gamma$ is a smooth, compact, oriented manifold. For an unconditional completion $\mathcal A(\Gamma,\sigma)$, write $\omega$ for the cohomology class associated with the multiplier and let $[\operatorname{tr}]$ denote the induced trace map on $K_0(\mathcal A(\Gamma,\sigma))$. In even dimension $2n$, choose generators $a_1(j),\ldots,a_{b_{2j}}(j)$ of $H^{2j}(B\Gamma,\mathbb Z)\cap H^{2j}(B\Gamma,\mathbb R)$, with $b_{2j}=\dim H^{2j}(B\Gamma,\mathbb R)$; in odd dimension $2n-1$, choose generators $a_1(j),\ldots,a_{b_{2j-1}}(j)$ of $H^{2j-1}(B\Gamma,\mathbb Z)\cap H^{2j-1}(B\Gamma,\mathbb R)$, with $b_{2j-1}=\dim H^{2j-1}(B\Gamma,\mathbb R)$. Trace-range conjecture. If $\dim B\Gamma=2n$, then

$$[\operatorname{tr}](K_0(\mathcal A(\Gamma,\sigma)))=\mathbb Z+\mathbb Z\theta+\sum_{j=1}^{n-1}\sum_{k=1}^{b_{2j}}\mathbb Z r_{k,j,n}\theta_k(j),$$

where $\theta_k(j)=\langle[\omega^{n-j}\cup a_k(j)],[B\Gamma]\rangle$, $2(2\pi)^n\theta=\langle[\omega^n],[B\Gamma]\rangle$, and $r_{k,j,n}$ are universal constants. If $\dim B\Gamma=2n-1$, then

$$[\operatorname{tr}](K_0(\mathcal A(\Gamma,\sigma)))=\mathbb Z+\sum_{j=1}^{n-1}\sum_{k=1}^{b_{2j-1}}\mathbb Z r'_{k,j,n}\theta_k(j),$$

where $\theta_k(j)=\langle[\omega^{n-j}\cup a_k(j)],[B\Gamma]\rangle$ and $r'_{k,j,n}$ are universal constants. The formulas extend the computed four-dimensional trace-range result to arbitrary even and odd dimensions, subject to the twisted Bost conjecture.