Greenberg's Iwasawa main conjecture for CM fields

Let ${\mathcal{K}}$ be a CM field with maximal totally real subfield ${\mathcal{F}}$, let $p$ be an odd prime unramified in ${\mathcal{F}}$ such that every prime above $p$ splits in ${\mathcal{K}}$, and let $W$ be the free part of the Galois group of the maximal pro-$p$ abelian extension of ${\mathcal{K}}$ unramified away from $p$. Put $\Lambda_W=\EuScript{O}\llbracket W\rrbracket$, let $\psi$ be a finite-order character of $\operatorname{\mathcal{G}}_{\mathcal{K}}$, and choose a CM type $\Sigma$ with associated set of primes $\Sigma_p$. For the Selmer group

$$\operatorname{Sel}(\psi,\Sigma_p)=\ker\big\{H^1({\mathcal{K}},\Lambda_W^*)\to\prod_{w\notin\Sigma_p}H^1(I_w,\Lambda_W^*)\big\},$$

write $X(\Lambda_W)=\operatorname{Sel}(\psi,\Sigma_p)^\vee$ and $F(\Lambda_W)=\operatorname{char}_{\Lambda_W}(X(\Lambda_W))$. Katz's $p$-adic $L$-function is denoted by $L_p(\psi,\Sigma_p)\in\Lambda_W$. Greenberg's Iwasawa main conjecture. The characteristic ideal $F(\Lambda_W)$ is generated by $L(\psi,\Sigma_p)$ in $\Lambda_W$. This conjecture identifies the characteristic ideal of the relevant Selmer module with the Katz--Hida--Tilouine $p$-adic $L$-function; the supplied text does not state whether it has been proved or disproved.