Binary finitely homogeneous trace-equivalence conjecture

A structure is binary finitely homogeneous if it admits quantifier elimination in a finite binary relational language. The trivial structure is the structure with no nontrivial relations; $(\mathbb{Q};<)$ is the rational dense linear order; and the ErdősRado graph is the corresponding homogeneous random graph. For a structure $\mathscr{O}$, let $\mathcal{C}_{\mathscr{O}}$ be the class of theories that do not trace define $\mathscr{O}$.

Binary trace-equivalence conjecture. Every structure admitting quantifier elimination in a finite binary relational language is trace equivalent to either the trivial structure, $(\mathbb{Q};<)$, or the theory of the ErdősRado graph. Equivalently, if $\mathscr{O}$ is binary finitely homogeneous, then $\mathcal{C}_{\mathscr{O}}$ is either the class of theories of finite structures, the class of stable theories, or the class of NIP theories.

The paper proves this in several important cases, including stable and IP theories and various classified homogeneous structures, but not in full generality.