In this paper, Gitik and Shelah make the following claim (part of Proposition 1.5):

Claim (Gitik-Shelah):Suppose $\kappa < \lambda$ are regular, $2^\lambda = \lambda^+$, and $D$ is a normal ideal on $\lambda$. If forcing with $D^+$ makes $cf(\lambda^+) < \kappa$, then it collapses all cardinals in some interval $(\alpha,\lambda^+]$ where $\alpha < \kappa$.

I do not understand the argument for this claim. An interesting dichotomy follows:

Corollary:If GCH holds and $\eta$ is a singular cardinal, then for every normal ideal $D$ on $\eta^+$, either $D$ is presaturated or forcing with $D$ collapses $\eta$.

Sketch of proof: Assume $D$ is not presaturated but $D^+$ preserves that $\eta$ is a cardinal. Since $D$ is not presaturated, forcing with $D^{+}$ collapses $\eta^{++}$, and since $|D^+| = \eta^{++}$, it preserves $\eta^{+3}$. By a well-known theorem of Shelah, this implies that $cf(\eta^{++}) = cf(\eta) < \eta$ in the extension by $D^+$. But by the Claim, $\eta$ is collapsed, a contradiction.

However, it seems that we should be able to force a counterexample to this dichotomy from large cardinals, but I am probably missing some subtlety. So my question is, how do you prove the Claim?