The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph_2$ or $\aleph_3$

We prove that the PFA lottery preparation of a strongly unfoldable cardinal $\kappa$ under $\neg 0^\sharp$ forces $\operatorname{PFA}(\aleph_2$-preserving$)$, $\operatorname{PFA}(\aleph_3$-preserving$)$ and $\operatorname{PFA}_{\aleph_2}$, with $2^\omega=\kappa=\aleph_2$. The method adapts to semi-proper forcing, giving $\operatorname{SPFA}(\aleph_2$-preserving$)$, $\operatorname{SPFA}(\aleph_3$-preserving$)$ and $\operatorname{SPFA}_{\aleph_2}$ from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction $\operatorname{SPFA}(\aleph_2$-preserving$) +\operatorname{SPFA}(\aleph_3$-preserving$)+ \operatorname{SPFA}_{\aleph_2}+2^\omega=\aleph_2$. Since unfoldable cardinals are relatively weak as large cardinal notions, our summary conclusion is that in order to extract significant strength from PFA or SPFA, one must collapse $\aleph_3$ to $\aleph_1$.