Countability and finite arity conjecture for finitely homogeneous structures modulo trace equivalence

A finitely homogeneous structure is a structure whose theory eliminates quantifiers in a finite relational language. Two theories are trace equivalent when each trace defines the other. The conjecture asserts that there are only countably many finitely homogeneous structures modulo trace equivalence, and, more strongly, for every $k65 1$ there are only finitely many theories admitting quantifier elimination in a finite $k$-ary relational language modulo trace equivalence.

Countability and finite arity conjecture. There are only countably many finitely homogeneous structures modulo trace equivalence. More strongly, for every $k65 1$ there are only finitely many theories admitting quantifier elimination in a finite $k$-ary relational language modulo trace equivalence.

This conjecture seeks a classification of finitely homogeneous structures up to trace equivalence; the paper establishes several substantial cases, including the binary setting, but the general finite-arity assertion remains open.