Greenberg's height-two annihilator conjecture for multiple $[200~\mathbb{Z}_p$-extensions

Let $p$ be a prime, let $K$ be a number field, and let $\widetilde{K}$ be the compositum of all $\mathbb{Z}_p$-extensions of $K$. Let $X$ be the Galois group of the maximal abelian unramified pro-$p$-extension of $\widetilde{K}$, and let $\Lambda$ denote the corresponding Iwasawa algebra. Greenberg's height-two annihilator conjecture. The annihilator of $X$ in $\Lambda$ has height at least $2$:

$$\operatorname{height}\bigl(\operatorname{Ann}_{\Lambda}(X)\bigr)\geq 2.$$

This generalizes the cyclotomic finiteness conjecture in the setting of all $\mathbb{Z}_p$-extensions; the supplied text does not state whether it has been resolved.