TITLE: Status of Harvey Friedman's grand conjecture?
QUESTION [34 upvotes]: Friedman [1] conjectured

Every theorem published in the Annals of Mathematics whose
  statement involves only finitary mathematical objects (i.e., what logicians
  call an arithmetical statement) can be proved in EFA. EFA is the weak
  fragment of Peano Arithmetic based on the usual quantifier free axioms for
  0,1,+,x,exp, together with the scheme of induction for all formulas in the
  language all of whose quantifiers are bounded. This has not even been
  carefully established for Peano Arithmetic. It is widely believed to be
  true for Peano Arithmetic, and I think that in every case where a logician
  has taken the time to learn the proofs, that logician also sees how to
  prove the theorem in Peano Arithmetic. However, there are some proofs which
  are very difficult to understand for all but a few people that have
  appeared in the Annals of Mathematics - e.g., Wiles' proof of FLT.

Have there been any serious challenges to this or the weaker conjecture with Peano arithmetic in place of exponential function arithmetic?
[1] http://cs.nyu.edu/pipermail/fom/1999-April/003014.html

REPLY [4 votes]: "Mathematische Annalen" and Annals of Mathematics aren't the same journal, but they have similar titles, so I'll try to slip this in.
Gentzen's proof of Con(PA) using transfinite induction on $\epsilon_0$ appeared in Mathematische Annalen in 1936 (cite pasted from Wikipedia):

G. Gentzen, 1936. 'Die Widerspruchfreiheit der reinen Zahlentheorie'. Mathematische Annalen, 112:493–565. Translated as 'The consistency of arithmetic', in (M. E. Szabo 1969). DOI: 10.1007/BF01565428, free eudml link.

Obviously this theorem can't be proved in PA itself, much less EFA.  I guess it doesn't really count but it seems sort of relevant.  It's surely a major theorem.