Van den Dries's transfer conjecture for o-minimal exponential fields

Let $\mathcal{K}=(K,+,-,\cdot,0,1,<,E)$ be an ordered exponential field, meaning that $E$ is an order-preserving isomorphism from the ordered additive group $(K,+,0,<)$ to the ordered multiplicative group $(K^{>0},\cdot,1,<)$. Call $\mathcal{K}$ an $\operatorname{EXP}$-field if its exponential satisfies the first-order sentence expressing the differential equation $E'=E$. Van den Dries's transfer conjecture. Any o-minimal $\operatorname{EXP}$-field is elementarily equivalent to

$$\mathbb{R}_{\exp}.$$

The conjecture would imply that every o-minimal $\operatorname{EXP}$-field has the complete first-order theory of real exponentiation. It is not known whether every o-minimal $\operatorname{EXP}$-field is already a model of $T_{\exp}$, so the conjecture remains open.