  For a prime number $p$ and a number field $k$, let $\tilde{k}$ be the
compositum of all $\mathbb{Z}_p$-extensions of $k$. Greenberg's Generalized
Conjecture (GGC) claims the pseudo-nullity of the unramified Iwasawa module
$X(\tilde{k})$ of $\tilde{k}$. It is known that, when $k$ is an imaginary
quadratic field, GGC has a consequence on the Iwasawa invariants associated to
$\mathbb{Z}_p$-extensions of $k$. In this paper, we partially generalize it to
arbitrary number fields $k$.
