On Greenberg's generalized conjecture for imaginary quartic fields

For an algebraic number field $K$ and a prime number $p$, let $\tilde{K}\mathrm{/}K$ be the maximal multiple ${\mathbb{Z}}_p$-extension. Greenberg’s generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-$p$ extension of $\tilde{K}$ is pseudo-null over the completed group ring ${\mathbb{Z}}_p[\text{\hspace{0.17em}⁣}[(\tilde{K}\mathrm{/}K)]\text{\hspace{0.17em}⁣}]$. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.