A consistency theorem for cardinal sequences of length $< \omega_3$

We prove that if $\lambda$ is a fixed uncountable cardinal and $f = \langle \ka_{\al} : \al < \delta \rangle$ is a sequence of infinite cardinals where $\delta < \omega_3$ and $\ka_{\al}\in \{\om,\lambda\}$ for each $\al < \delta$ in such a way that $f^{-1}\{\om\}$ is $\om_2$-closed in $\delta$, then it is consistent that there is a scattered Boolean space whose cardinal sequence is $f$.