Weak Greenberg's generalized conjecture for maximal multiple $\p$-extensions

Let $p$ be a prime number and $k$ a number field. Let $\widetilde{k}$ be the maximal multiple $\mathbb{Z}_p$-extension field of $k$, let $L_{\widetilde{k}}$ be its maximal unramified pro-$p$ abelian extension field, and write

$$X_{\widetilde{k}}=\operatorname{Gal}(L_{\widetilde{k}}/\widetilde{k}).$$

A $\mathbb{Z}_p[[\operatorname{Gal}(\widetilde{k}/k)]]$-submodule is pseudo-null if its annihilator has height greater than one. Weak Greenberg's generalized conjecture. Let $p$ be a prime number and $k$ a number field. If $X_{\widetilde{k}}$ is not trivial, then $X_{\widetilde{k}}$ has a non-trivial pseudo-null $\mathbb{Z}_p[[\operatorname{Gal}(\widetilde{k}/k)]]$-submodule. This weak form asks only for a nonzero pseudo-null submodule of the Iwasawa module, and the paper proves it for certain imaginary quadratic fields under additional vanishing and square-free characteristic-ideal hypotheses; it remains open in general.