For a countable ordinal $\eta$ and an ordinal $\gamma$ let $\langle P_\alpha, \dot{Q}_\alpha \colon \alpha < \gamma \rangle$ be an RCS iteration with RCS limit $P_\gamma$, such that

• $\Vdash_\alpha \dot{Q}_\alpha$ is $\eta$-semi-proper;
• $\Vdash_{\alpha +1} \vert P_\alpha \vert \leqslant \aleph_1$.

Is then the RCS limit $P_\gamma$ also semi-proper?

For a countable ordinal $\eta$ and an ordinal $\gamma$ let $\langle P_\alpha, \dot{Q}_\alpha \colon \alpha < \gamma \rangle$ be an RCS iteration with RCS limit $P_\gamma$, such that

• $\Vdash_\alpha \dot{Q}_\alpha$ is $\eta$-semi-proper;
• $\Vdash_{\alpha +1} \vert P_\alpha \vert \leqslant \aleph_1$.

Is then the RCS limit $P_\gamma$ also $\eta$-semi-proper?