Thomas's finite reducts conjecture for finite-language homogeneous structures

Let $\Delta$ be a countable relational structure that is homogeneous in a finite language. A reduct of $\Delta$ is a relational structure on the same domain whose relations are first-order definable in $\Delta$. Two structures are first-order interdefinable when each is a reduct of the other.

Thomas's conjecture. The structure $\Delta$ has only finitely many reducts up to first-order interdefinability.

This conjecture predicts finiteness for the reduct classification of highly symmetric homogeneous structures. It generalizes known classifications of several prominent $\omega$-categorical structures, but the source provides no resolution of the general case.