Greenberg's pseudo-nullity conjecture

Let $F$ be a number field, let $\widetilde{F}$ be the compositum of all $\mathbb{Z}_{p}$-extensions of $F$, and let $K(\widetilde{F})$ be the maximal unramified abelian pro-$p$ extension of $\widetilde{F}$ in which every prime above $p$ splits completely. Greenberg's conjecture. The group

$$\operatorname{Gal}(K(\widetilde{F})/\widetilde{F})$$

is a pseudo-null $\mathbb{Z}_{p}[[\operatorname{Gal}(\widetilde{F}/F)]]$-module. This is a central structural conjecture in multivariable Iwasawa theory; the supplied source does not state whether it has been resolved.