The restrained-structure dichotomy conjecture

A restrained structure is a DC expansion of an ordered field such that, for every definable discrete set $D \subseteq \mathbb K^n$ and every definable function $g:D\to\mathbb K$, the image $g(D)$ is nowhere dense in $\mathbb K$. A structure defines a discrete subring if it defines a subring of $\mathbb K$ containing $1$ whose underlying set is discrete. The restrained-structure dichotomy conjecture. Either $\mathbb K$ is restrained, or it defines a discrete subring containing $1$. The claim proposes a dichotomy between restrained tame behavior and the presence of a discrete arithmetic-like subring; no resolution is supplied in the source.