The countability conjecture for finitely homogeneous structures

A finitely homogeneous structure is a structure whose age is a finitely homogeneous Fraïssé class; structures are considered up to trace equivalence.

Countability conjecture. There are $\aleph_0$ finitely homogeneous structures up to trace equivalence.

The paper gives evidence for infinitude by separating generic hypergraphs of different arities. It then proposes a stronger finite-per-arity conjecture, so this countability assertion remains part of the paper's conjectural classification.