Can anyone suggest a reason to believe or disbelieve the following assertion:

There is a theory $T$ extending open induction such that

1. $T$ has a recursively presentable nonstandard model.

2. If the sentence $\phi$ is not provable from $T$, then $T+\phi$ has no recursively presentable nonstandard model.

This is version 2 of a question about the ultimate limits of Tennenbaum's Theorem. The attempt to find these limits by moving up the induction heirarchy, as in Wilmer's Theorem, seems somehow indecisive. I suggested that maybe there is a Theory $T$ extending open induction such that

1. $T$ has a recursively presentable nonstandard model.

2. If the sentence $\phi$ is not provable from $T$, then $T+\phi$ has no recursively presentable nonstandard model.

François G. Dorais immediately replied that this just amounts to $T$ being complete.

So... What about asking for the maximum $n$ such that the theory of all true (in the integers) all-2 sentences with n existential quantifiers has a recursive nonstandard model? What is known about this? Is it known that $n<2$?