Friedman's 14-th problem.
From Peano's Parlour
Let T be a completion of PA. Let Ot(T) be the spectrum of order types of nonstandard models of T. In [1] Harvey Friedman asked: Does Ot(T) depend on T?
Shelah has recently announced that the answer in negative, but there is no paper available yet. See Sh[924] for an interesting effort in the opposite direction.