Let $K$ be an imaginary quadratic field and $\widetilde{K}$ be the compositum of all $\mathbb{Z}_p$ extensions of $K$. Here, $\widetilde{K}/K$ is a $\mathbb{Z}_p^2$ extension.

Define $M(\widetilde{K})$ to be the maximal Abelian unramified pro-$p$ extension of $\widetilde{K}$.

The generalized Greenberg's conjecture (GGC) says the Greenberg-Iwasawa module, $X_{\widetilde{K}}\simeq \textrm{Gal}(M(\widetilde{K})/\widetilde{K})$ is pseudo-null.

[There are lots of examples where this conjecture is known.]

Let $F/K$ be any finite extension of $K$, and consider the compositum $F\widetilde{K}$. This is a $\mathbb{Z}_p^2$ extension over $F$. Are there any (partial) results that prove $X_{F\widetilde{K}}$ is pseudo-null when GGC for $\widetilde{K}$ is known?