Friedman's 14-th problem.

Let $T$ be a completion of PA. Let ${\rm Ot}(T)$ be the spectrum of order types of nonstandard models of $T$.

Does ${\rm Ot(T)}$ depend on $T$?

Friedman, Harvey, One hundred and two problems in mathematical logic. J. Symbolic Logic 40 (1975), 113–129.

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.