Greenberg's conjecture for arbitrary Galois extensions

Let $bK/Q$ be an arbitrary Galois extension containing the $p$-th roots of unity, and let $H/K_{\infty}$ be the maximal unramified $p$-abelian extension of the cyclotomic $Z_p$-extension $K_{\infty}$ of $K$. Define

$$\Omega_E=\bigcup_n K_n[E_n^{1/p^n}],\qquad E_n=O(K_n),$$

and set $H_{E'}=H\cap\Omega_E$. Greenberg's conjecture.

$$[H:H_{E'}]<\infty.$$

This is presented as a general formulation of Greenberg's conjecture, motivated by numerical evidence and by equivalent or related finiteness assertions such as $\lambda^+=0$ and finiteness of $A^+$ for CM fields. The supplied text gives no resolution, so the conjecture remains open in this generality.