Ordered abelian groups and projective planes conjecture

An ordered abelian group is an abelian group equipped with a compatible linear order. A theory locally trace defines a structure when it does so at the paper's local trace-definability level. An infinite projective plane is a projective plane with infinitely many points and lines.

Ordered-group projective-plane conjecture. An ordered abelian group cannot locally trace define an infinite projective plane.

The same paragraph also states that an ordered abelian group cannot locally trace define an infinite field as an assertion, while presenting only the projective-plane claim as the additional conjectural ambition. The supplied text gives no resolution.