Reference request:  Forcing Axiom for the class of Axiom A posets Does anyone know of a paper which asks, or conjectures about, the following question?
QUESTION:  Is the Forcing Axiom for Axiom A posets equiconsistent with the Proper Forcing Axiom (PFA)?
Axiom A posets are a subclass of the proper posets, defined by Baumgartner around the same time as the discovery of proper forcing by Shelah.  Many consequences of PFA (e.g. those involving finite iterations of c.c.c. and $\sigma$-closed posets) are consequences of the Forcing Axiom for Axiom A.
 A: this may help (building on earlier work by Ishiu and Weinert):
Bounded forcing axioms and Baumgartner's conjecture, by Aspero, Friedman, Mota & Sabok.

We study the spectrum of forcing
  notions between the iterations of
  σ-closed followed by ccc forcings and
  the proper forcings. This includes the
  hierarchy of α-proper forcings for
  indecomposable countable ordinals α as
  well as the Axiom A forcings. We focus
  on the bounded forcing axioms for the
  hierarchy of α-proper forcings and
  connect them to a hierarchy of weak
  club guessing principles. We show that
  they are, in a sense, dual to each
  other. In particular, these weak club
  guessing principles separate the
  bounded forcing axioms for distinct
  countable indecomposable ordinals. In
  the study of forcings completely
  embeddable into an iteration of
  σ-closed followed by ccc forcing, we
  present an equivalent characterization
  of this class in terms of
  Baumgartner’s Axiom A. This resolves a
  well-known conjecture of Baumgartner
  from the 1980’s.

