Say that a complete theory $T$ is a limit of stable theories if for every $\phi \in T$ there is a stable completion of $\{\phi\}$. (Equivalently, $T$ is the ultraproduct of stable theories.)

**Question:** Is every simple theory a limit of stable theories? (We allow that theories of finite structures are stable.)

As motivation, all the simple unstable theories I am aware of are random in various ways (e.g. random graph) and the independence property only comes from the full random schema.

Example: any pseudo-finite theory is a limit of stable theories. This includes a lot of of simple theories, e.g. the theory of the random graph. It also includes pseudo-finite linear orders, which are SOP.

Example: suppose $T$ is a completion of ACFA (the model companion of algebraically closed fields with an automorphism), so $T$ is simple. Suppose $\phi \in T$. Then $\phi$ is consistent with ACFA, so for arbitrarily large powers of primes $p^n$, $(F, x \mapsto x^{p^n}) \models \phi$, where $F$ is some (any) algebraically closed field of characteristic $p$. But these structures are stable.

A strong counterexample would be a finitely axiomatizable simple unstable theory.