Finite-arity conjecture for finitely homogeneous structures modulo trace equivalence

A $d$-ary finitely homogeneous structure is a finitely homogeneous structure presented in a relational language whose relation symbols have arity at most $d$. Two such structures are trace equivalent when each trace defines the other.

Finite-arity conjecture. For every $d\in\mathbb{N}_{\ge 1}$, there are only finitely many $d$-ary finitely homogeneous structures modulo trace equivalence.

The binary case has countably many classes by the results discussed in the paper, while the asserted finiteness for every fixed arity remains open.