TITLE: Properness for small forcings
QUESTION [8 upvotes]: It is easy to see a forcing of size $\aleph_1$ is proper if and only if is semiproper. I was wondering when such an equivalency holds between semi-proper and stationary-preserving forcings in $\rm ZFC$? Or consistently in a model where significant fragments of $\rm MM$ fail.

REPLY [3 votes]: The answer is no, as the following upcoming work of Shelah and Usuba shows:
Theorem (Shelah-Usuba):
The following theories are equiconsistent with ZFC:
ZFC+CH+ “there is an $\omega _1$-stationary preserving $\sigma$-Baire poset of size $\aleph_1$
​    which is not semiproper”.
ZFC+“Martin’s axiom for semiproper posets of size $\aleph_1$” + “there is an $\omega _1$-stationary preserving $\sigma$-Baire poset of size $\aleph_1$ which is not semiproper”.
ZFC+CH+“every $\omega _1$-stationary preserving $\sigma$-Baire poset of size $\aleph_1$ is semiproper”.
See $\omega_1$-Stationary preserving $\sigma$-Baire posets of size $\aleph_1$.