Baumgartner's conjecture and bounded forcing axioms

We study the spectrum of forcing notions between the iterations of s-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of a-proper forcings for indecomposable countable ordinals a, the Axiom A forcings and forcings completely embeddable into an iteration of a s-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartner's Axiom A. This resolves a conjecture of Baumgartner from the 1980s. We also study the bounded forcing axioms for the hierarchy of a-proper forcings. Following ideas of Shelah we separate them for distinct countable indecomposable ordinals