Twisted Bost conjecture for unconditional twisted group-algebra completions

Let $\Gamma$ be a countable group and let $\sigma$ be a multiplier on $\Gamma$ with trivial Dixmier–Douady invariant. Let $\mathcal A(\Gamma,\sigma)$ be any unconditional completion of the twisted group algebra $\mathbb C(\Gamma,\sigma)$, and let $\mu_\sigma^{\mathcal A}$ denote the twisted assembly map

$$\mu_\sigma^{\mathcal A}:K_j^\Gamma(\underline E\Gamma)\longrightarrow K_j(\mathcal A(\Gamma,\sigma)),\qquad j=0,1.$$

Twisted Bost conjecture. The map $\mu_\sigma^{\mathcal A}$ is an isomorphism for $j=0,1$. The conjecture is presented as an unconditional analogue of the twisted Baum–Connes conjecture and as a twisted Bost conjecture when the completion is $\ell^1$; the source gives no resolution status.