Let $M_1$ be the canonical inner model with one Woodin cardinal $\delta$. Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic filter for the forcing. Note that, assuming $M_1^{\sharp}$ exists, there exists a formula which defines $M_1$ in the generic extension properly. Suppose that in $M_1[G]$ the $(M_1)$-initial segment $(M_1)_{\alpha}$ has size $\aleph_1$.

My question is whether it is possible, for unboundedly many $\alpha < \aleph_2^{M_1[G]}$ to find an elementary continuous chain $(N_i \, : \, i \in \omega_1)$ of elementary submodels of $(M_1)_{\alpha}$ such that $\bigcup_{i < \omega_1} N_i = (M_1)_{\alpha}$, and such that for every model $N_i$ in the chain, its transitive collapse is an initial segment of $M_1$ again?

Note that the statement is true if $\mathbb{P}$ is just the trivial forcing, however I would be interested in the case when we collapse cardinals down to $\omega_1$.

2more comments