On generlized Ehrenfeucht-Mostowski models

One of the main question in generalized descriptive set theory is whether there is generalized Borel-reducibility counterpart of Shelah's main gap theorem.
Which is stated as follows:
Let $T_1$ be a classifiable theory and $T_2$ a non-classifiable theory. Is there a Borel reduction from the isomorphism relation of $T_1$ to the isomorphism relation of $T_2$?

This question was studied by Friedman, Hyttinen, and Weinstein (former Kulikov), who gave a positive answer to this question for the case $\kappa$ a successor cardinal (under certain cardinal assumptions), and $T_2$ a stable unsuperstable theory.
All of the above assume the existence of diamond on a stationary set in order to execute the proof.
The case when $\kappa$ is strongly-inaccessible cardinal was study by Hyttinen and Moreno.

In a joint paper with Moreno, a positive answer to this question is given for $\kappa$ an inaccessible cardinal, $T_2$ a stable theory with OCP.

In this talk we give a brief overview of the subject and present new result which extends to unsuperstable theories for strongly inaccessible cardinal.
Thus, in particular giving a positive answer to the question without the use of diamond.