Equivariant Iwasawa main conjecture for Tate motives

Let $k_\infty/k$ be a $\mathds Z_p$-power extension with no finite place splitting completely, let $K_\infty=Kk_\infty$, and use the source's decomposition and character $\chi$ to define $T_\chi=\mathcal O(1)(\chi)$ and $\mathcal T_\chi=\Lambda_F(1)(\chi)$. Let $\Theta_{K_\infty/k,S(K),b_\bullet}$ be the projection map and let $\varepsilon^\chi_{K_\infty,\Sigma}$ be the induced Rubin--Stark Euler-system element. Equivariant Iwasawa main conjecture. There is a $\Lambda_F$-basis $\mathfrak z^\chi_{F_\infty,S(K),\Sigma}$ of $\operatorname{Det}_{\Lambda_F}(C_{S(K),\Sigma}(\mathcal T_\chi))$ such that

$$\varepsilon^\chi_{K_\infty,\Sigma}=\Theta_{K_\infty/k,S(K),b_\bullet}(\mathfrak z^\chi_{K_\infty,S(K),\Sigma}).$$

This asserts that the Rubin--Stark Euler-system element generates the determinant line under the projection map, giving an equivariant higher-rank Iwasawa main conjecture; no resolution is supplied in the source.