Pouzet's conjecture on totally ordering omega-well structured relational structures

Let $\mathcal{A}$ be a relational structure. It is $\omega$-well structured if it satisfies the relevant well-structuredness condition for embeddings, and it is a reduct of another relational structure when the latter has the same underlying set with possibly additional relations. A relational structure is totally ordered when its language includes a total order.

Pouzet's conjecture. Every $\omega$-well structured relational structure $\mathcal{A}$ is a reduct of an $\omega$-well structured, totally ordered structure.

The conjecture is presented as asserting that the necessary and sufficient conditions in the paper's equivariant Noetherianity theorems coincide. Its resolution is not specified in the supplied text.