Higher-rank non-commutative main conjecture for \mathbb{G}_m

Let $\mathcal{K}_\infty/K$ and $S$ be as in the source, put $\Sigma=\Sigma_S(\mathcal{K}_\infty)$ and $r=|\Sigma|$, and let $\Lambda(\mathcal{G}_\infty)$ be the relevant Iwasawa algebra with total quotient ring $Q(\mathcal{G}_\infty)$. Let $C_{\mathcal{K}_\infty,S,T}$ be the inverse-limit cohomological complex, $\varepsilon^{\rm RS}_{\mathcal{K}_\infty,S,T}$ the inverse-limit Rubin–Stark element, and $\Theta^\Sigma_{\mathcal{K}_\infty,S,T}$ the canonical homomorphism to the rank-$r$ bidual module. Higher-rank non-commutative main conjecture for $\mathbb{G}_m$. One has

$${\rm Nrd}_{Q(\mathcal{G}_\infty)}(K_1(\Lambda(\mathcal{G}_\infty)))\cdot\varepsilon^{\rm RS}_{\mathcal{K}_\infty,S,T}=\Theta^\Sigma_{\mathcal{K}_\infty,S,T}({\rm d}_{\Lambda(\mathcal{G}_\infty)}(C_{\mathcal{K}_\infty,S,T})^{\rm pb})$$

in ${\bigcap}_{\Lambda(\mathcal{G}_\infty)}^r\mathcal{O}_{\mathcal{K}_\infty,S,T}^\times$. This is an explicit higher-rank main conjecture in non-commutative $p$-adic Iwasawa theory for $\mathbb{G}_m$; the supplied text gives no resolution status.