Bi-interpretability with classes covered by the completion method

Construct, for the limit of antilexicographically ordered finite Boolean algebras, convexly ordered finite C-relations, or finite meet-semilattices with a linear extension, a first-order bi-interpretable homogeneous structure whose age is a locally finite subclass of finite linearly ordered structures.

Background

The question asks whether Ramsey classes whose known proofs use dual Ramsey or tree methods can be transformed into structures amenable to the general completion theorem for Ramsey classes.

References

Is there a language $L$ containing $<$, and a homogeneous $L$-structure $\str H$ which is first-order bi-interpretable with the limit of $BA+$, or the class of all finite convexly ordered $C$-relations, or the class of all finite meet-semilattices with a linear extension, such that $(\str H)$ is a locally finite subclass of the class of all finite linearly ordered $L$-structures?

— Twenty years of Nešetřil's classification programme of Ramsey classes  (2501.17293 - Hubička et al., 28 Jan 2025) in Question, subsection Ramsey classes