The smoothed equivariant $p$-adic Artin conjecture with uniqueness

Let $L/K$ be as above, and let $S$ and $T$ be finite non-empty disjoint sets of places of $K$, with $S$ containing all places ramifying in $L^+_\infty/K$ and all infinite places. Write $\mathcal G$ for the relevant Galois group, $\Lambda(\mathcal G)$ for its Iwasawa algebra, $\mathcal Q(\mathcal G)$ for its total quotient algebra, $\operatorname{nr}$ for the reduced norm, and $\partial$ for the boundary homomorphism in relative algebraic $K$-theory. Let $\Phi_S^T$ be the smoothed equivariant $p$-adic $L$-function and let $[Y_S^T]\in K_0(\Lambda(\mathcal G),\mathcal Q(\mathcal G))$ be the indicated arithmetic class. Smoothed equivariant $p$-adic Artin conjecture with uniqueness. There is a unique element $\zeta_S^T(L^+_\infty/K)\in K_1(\mathcal Q(\mathcal G))$ such that

$$\operatorname{nr}\left(\zeta_S^T(L^+_\infty/K)\right)=\Phi_S^T.$$

Moreover,

$$\partial\left(\zeta_S^T(L^+_\infty/K)\right)=[Y_S^T].$$

This is the smoothed version of the equivariant Iwasawa main conjecture for the $p$-adic Artin $L$-function. The preceding discussion identifies the relevant complex with the class $[Y_S^T]$ and explains the smoothing factors; the uniqueness assertion is equivalent to the vanishing of $S\!K_1(\mathcal Q(\mathcal G))=\ker(\operatorname{nr})$.