The classical Namba forcing collapses $\omega_2$ to have cofinality $\omega$ while preserving the cardinal $\omega_1$. Higher analogs were constructed to (under some additional hypotheses) give a cardinal $\omega_n$ cofinality $\omega$ while preserving $\omega_k$ for $k<n$.

However, i am interested in a different generalisation which collapses to $\omega_1$ instead of $\omega$. So: Does there (consistently) exist a forcing notion that forces $\omega_3$ to have cofinality $\omega_1$ while preserving $\omega_2$?