O-minimal trace-definable field conjecture

An o-minimal structure is a structure expanding a linear order in which every definable subset of the underlying line is a finite union of points and intervals. A field is trace definable in an o-minimal structure if its structure is obtained by the paper's trace-definability notion from an o-minimal structure. A field is real closed in the usual ordered-field sense, and has characteristic zero when its prime field is isomorphic to $\mathbb{Q}$.

O-minimal field conjecture. Any infinite field trace definable in an o-minimal structure is real closed or algebraically closed of characteristic zero.

The conjecture is proposed as a target for applying finite-dp-rank field classification; the supplied text does not state a proof or disproof.