The ordered-field conjecture for special relativity with approximate expansion

Let $d\ge2$. Write $\operatorname{Num}(\mathsf{Th})$ for the class of quantity structures of models of a theory $\mathsf{Th}$, and let $\mathsf{SpecRel}_d$ be the axiom system for $d$-dimensional special relativity. Let $\mathsf{AxThExp}^{-}$ be the approximate version of the axiom asserting that inertial observers can move with arbitrary subluminal speeds. An ordered field is a field equipped with a compatible total order. The ordered-field conjecture. For all $d\ge2$,

$$\operatorname{Num}(\mathsf{SpecRel}_d+\mathsf{AxThExp}^{-})=\{\mathfrak{Q}:\mathfrak{Q}\text{ is an ordered field}\}.$$

The question of exactly which ordered fields occur as quantity structures is stated to be open; the conjecture would characterize them all, in every dimension, extending the established rational and Archimedean-field cases.