The abelian Iwasawa main conjecture for CM fields

Let $F$ be a CM field and let $\rho\colon G_F\to\operatorname{Aut}_{\mathcal{E}}(V_\rho)$ be an Artin representation such that the field $F_\rho$ corresponding to the kernel of $\rho$ is linearly disjoint from $F_{\max}$. Let $\mathcal{O}$ be the ring of integers of $\mathcal{E}$. Choose a $G_F$-stable $\mathcal{O}$-lattice $T$ of $V_\rho$ and set $A=T\otimes_{\mathbb{Z}_p}\mathbb{Q}_p/\mathbb{Z}_p$. Let $\operatorname{Sel}_{A,\Sigma_F}(F_{\max})$ be the Selmer group defined above.

Iwasawa main conjecture for $\psi$. In the abelian case, for a finite Hecke character $\psi^{\mathrm{gal}}\colon G_F\to\operatorname{Aut}_{\mathcal{E}}V_\psi$ with $F_\psi$ linearly disjoint from $F_{\max}$, the equality

$$(L_{p,\Sigma_F}(\psi))=\operatorname{char}_{\widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F,\max}]]}\bigl(\operatorname{Sel}_{A_\psi,\Sigma_F}(F_{\max})^\vee\widehat{\otimes}_{\mathcal{O}}\widehat{\mathcal{O}}^{\mathrm{ur}}\bigr)$$

of principal ideals of $\widehat{\mathcal{O}}^{\mathrm{ur}}[[\Gamma_{F,\max}]]$ should hold, where $L_{p,\Sigma_F}(\psi)$ is the $p$-adic Hecke $L$-function defined as in the cited theorem. The Pontryagin dual of the Selmer group is pseudo-isomorphic to the corresponding Iwasawa module, and this conjecture is equivalent to the main conjecture proposed by Hida and Tilouine.