A note on adding isomorphisms and the pseudointersection number

We prove that for every tower $\mathcal{T}$ there are ${\mathrm{\aleph }}_1$-dense $A$ and $B$ so that any ``reasonable" forcing notion $\mathbb{P}$ – an adjective that includes all known ones – for making $A$ and $B$ isomorphic will add a pseudointersection for the tower. This shows in particular that ${\mathsf{M}\mathsf{A}}_{{\mathrm{\aleph }}_1}(\sigma -\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{d})$ holds in all known models of $\mathsf{B}\mathsf{A}$, which provides intrigue to well known questions of Todorčević and Steprāns-Watson.