# Complexity and classification of countable models

Let $T$ be a completion of $PA$. It is not hard to see that the isomorphism problem for finitely generated models of $T$, $\cong^{fg}_T$, is Borel.
Coskey and Kossak proved that $\cong^{fg}_T$, is essentially countable and $E_0\leq_B \cong^{fg}_T$ i.e. $\cong^{fg}_T$ is not smooth. Is $\cong^{fg}_T$ hyperfinite? In other words, is $\cong^{fg}_T$ Borel reducible to $E_0$?