Greenberg's conjecture and capitulation in Z_p^d-extensions

Let p be an odd prime. Let k be an algebraic number field and let \tilde{k} be the compositum of all the \mathbb{Z}_p-extensions of k, so that Gal(\tilde{k}/k) \simeq \mathbb{Z}_p^d for some finite d. We shall consider fields k with $Gal(k/\mathbb{Q}) \simeq (\mathbb{Z}/2\mathbb{Z})^n . Building on known results for quadratic fields, we shall show that the Galois group of the maximal abelian unramified pro-p-extension of \tilde{k} is pseudo-null for several such k's, thus confirming a conjecture of Greenberg. Moreover we shall see that pseudo-nullity can be achieved quite early, namely in a \mathbb{Z}_p^2-extension, and explain the consequences of this on the capitulation of ideals in such extensions.