On Namba Forcing and Minimal Collapses

We build on a 1990 paper of Bukovsky and Coplakova-Hartova. First, we remove the hypothesis of $\mathsf{\text{CH}}$ from one of their minimality results. Then, using a measurable cardinal, we show that there is a $\mathrm{\mid }{\mathrm{\aleph }}_2^V\mathrm{\mid }={\mathrm{\aleph }}_1$-minimal extension that is not a $\mathrm{\mid }{\mathrm{\aleph }}_3^V\mathrm{\mid }={\mathrm{\aleph }}_1$-extension, answering the first of their questions.