Weak Greenberg's generalized conjecture for imaginary quadratic fields

Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for $p$ and $k$, which states that the non-trivial Iwasawa module of the maximal multiple $\mathbb{Z}_p$-extension field over $k$ has a non-trivial pseudo-null submodule. We prove this conjecture for $p$ and $k$ under the assumption that the Iwasawa $\lambda$-invariant for a certain $\mathbb{Z}_p$-extension over a finite abelian extension of $k$ vanishes and that the characteristic ideal of the Iwasawa module associated to the cyclotomic $\mathbb{Z}_p$-extension over $k$ has a square-free generator.