The ordered-field IP conjecture on the natural numbers

Let $\oplus$ and $\otimes$ be binary operations on $\mathbb{N}$ such that $(\mathbb{N},\oplus,\otimes)$ is a field. Ordered-field IP conjecture. The expansion $(\mathbb{N},<,\oplus,\otimes)$ has the independence property. This conjecture is presented as a special case of the directed-family conjecture and would obstruct tame expansions interpreting an infinite field on a discrete domain.