Noncommutative Iwasawa main conjecture for the Bloch–Kato Selmer group

Let $V\cong\mathcal K^{2d}$ be an even-dimensional $p$-adic representation of $G_{\mathbb Q}$ satisfying condition $\mathrm{(Geom)}$, with no Artin representation as a subquotient. Let $\mathcal O$ be the ring of integers of a finite extension of $\mathbb Q_p$, let $G=\operatorname{Gal}(K_\infty/\mathbb Q)$, and let $H$ be the subgroup used to define $\mathfrak M_H(G)$. Assume the Hasse–Weil conjecture, Deligne's algebraicity conjecture, and the conjectural existence of the analytic $p$-adic $L$-function. Let $\mathcal L_p(V)\in K_1(\Lambda_{\mathcal O}(G)_{S^*})$ be that $p$-adic $L$-function, and let $\delta$ be the boundary map to $K_0(\mathfrak M_H(G))$. Iwasawa Main Conjecture. One has

$$\delta(\mathcal L_p(V))=[\operatorname{Sel}^{\mathrm{BK}}_A(K_\infty)^\vee]\in K_0(\mathfrak M_H(G)).$$

This is the $K$-theoretic formulation relating the analytic $p$-adic $L$-function to the Bloch–Kato Selmer group. It is stated conditionally on the preceding conjectures, and the supplied text gives no resolution.