The dichotomy conjecture for conjugacy problems of automorphism groups

Let $cmathcal M$ be a countable model of an $caleph_0$-categorical theory, let $cmathrm{aut}cmathcal M$ be its automorphism group, and let $cmathcal C_{cmathcal M}$ be the conjugacy equivalence relation on $cmathrm{aut}cmathcal M$, defined by

$$f\mathrel{\mathcal C_{\mathcal M}}g\iff(\exists h\in\operatorname{aut}\mathcal M)(hf=gh).$$

Conjugacy-complexity dichotomy conjecture. If $cmathcal M$ is the countable model in an $caleph_0$-categorical theory, then $cmathcal C_{cmathcal M}$ is either smooth or Borel complete.

This conjecture asks which Borel complexities can occur for conjugacy relations of automorphism groups of countable models of $caleph_0$-categorical theories. The paper establishes Borel completeness for the conjugacy problem of the random graph, while noting smooth examples such as the automorphism group of a countable set and that of the complete binary tree; the general dichotomy remains open in the supplied text.