The local-global cohomological conjecture for relative K-groups

Let $F$ be a number field and let $G$ be a finite group on which $\Omega_F$ acts trivially. Define

$$\Psi:H^1_t(F,G)\to K_0(O_FG,F^c),\qquad [\pi]\mapsto[O_\pi,O_FG;{\mathbf r}_G],$$

and set $K\mathcal R(O_FG)=\operatorname{Im}(\Psi)$. For each finite place $v$ of $F$, define the local map $\Psi_v$ analogously, and call an element of $K_0(O_FG,F^c)$ cohomological if it lies in $\operatorname{Im}(\Psi)$ and locally cohomological if its localisation lies in $\operatorname{Im}(\Psi_v)$ for every finite place $v$. Let

$$\lambda:K_0(O_FG,F^c)\to J(K_0(O_FG,F^c))$$

be the injective localisation map, and let $\Psi^{id}$ be the induced idelic map. The local-global cohomological conjecture. An element of $K_0(O_FG,F^c)$ is cohomological if and only if it is locally cohomological; equivalently,

$$\lambda(K\mathcal R(O_FG))=\operatorname{Im}(\lambda)\cap\operatorname{Im}(\Psi^{id}).$$

This conjecture gives a local-global characterisation of the image of the global resolvend map in relative algebraic $K$-theory. The source does not state a resolution; the result is open in this record.