Greenberg's generalized conjecture for maximal multiple extensions

Let $k/\mathbb{Q}$ be a finite extension and $p$ a prime number. Let $\widetilde{k}/k$ be the maximal multiple $\mathbb{Z}_p$-extension, and let $X(\widetilde{k})$ be its unramified Iwasawa module. A module is pseudo-null over $\mathbb{Z}_p\llbracket\operatorname{Gal}(\widetilde{k}/k)\rrbracket$ if it is pseudo-null in the usual Iwasawa-theoretic sense.

Greenberg's generalized conjecture. The module $X(\widetilde{k})$ is pseudo-null as a $\mathbb{Z}_p\llbracket\operatorname{Gal}(\widetilde{k}/k)\rrbracket$-module.

This generalizes Greenberg's conjecture from the cyclotomic extension to the maximal multiple $\mathbb{Z}_p$-extension and is open in general.