The equivariant local Iwasawa main conjecture for unramified extensions

Let $L/K$ be a finite Galois extension of $p$-adic local fields, let $L_\infty$ be the unramified $\mathbb{Z}_p$-extension of $L$, and let $\mathcal{G}=\operatorname{Gal}(L_\infty/K)$. Choose an isomorphism $j:\mathbb{C}\simeq\mathbb{C}_p$. Let $\mathcal{Q}^c(\mathcal{G})$ be the relevant total quotient algebra, let $K_1$ and $K_0$ denote the corresponding algebraic $K$-groups, let $C_{L_\infty/K}$, $U'_{L_\infty/K}$, $M_{L_\infty/K}$, and $\tau^{(j)}_{L_\infty/K}$ be the cohomological, unramified, arithmetic, and epsilon-constant terms defined in the paper, and let $\partial_{\Lambda(\mathcal{G}),\mathcal{Q}^c(\mathcal{G})}$ and $\operatorname{Det}$ be the connecting homomorphism and determinant map.

The equivariant local Iwasawa main conjecture. There exists $\zeta^{(j)}_{L_\infty/K}\in K_1(\mathcal{Q}^c(\mathcal{G}))$ such that

$$\partial_{\Lambda(\mathcal{G}),\mathcal{Q}^c(\mathcal{G})}(\zeta^{(j)}_{L_\infty/K})=-C_{L_\infty/K}-U'_{L_\infty/K}+M_{L_\infty/K}$$

and

$$\operatorname{Det}(\zeta^{(j)}_{L_\infty/K})=\tau^{(j)}_{L_\infty/K}.$$

This is the main conjecture formulated in the paper; the authors give strong evidence and prove it in the tamely ramified case, while the general assertion remains open.