The generalized non-commutative main conjecture for Tate motives

Let $K\in\mathcal{F}^+$ be unramified outside a finite set of places $\Sigma$, and let $C_K$ be the associated compactly supported arithmetic complex. Generalized Tate-motive main conjecture. The complex $C_K$ belongs to $D^{\rm p}_{S^*}(\Lambda(\mathcal{G}_K))$, and there exists $\xi'\in K_1(\Lambda(\mathcal{G})_{S^*})$ such that, for every Artin character $\rho$ of $G$, $\xi'(\rho)=\Omega_j(\rho)L_\Sigma^*(1,\rho)^j$, and $\partial_\mathcal{G}(\xi')=[C_K]$. This formulation is intended to extend the earlier Tate-motive main conjecture to groups that may contain elements of order $p$.