The generic smooth expansion conjecture for semialgebraic Mordell–Lang circle groups

Let $I$ be a closed bounded interval with nonempty interior, and let $C^\infty(I)$ be the topological vector space of smooth functions $I\to\mathbb{R}$. Let $\mathbb{H}$ be a semialgebraic Mordell–Lang circle group, let $\gamma:\mathbb{R}/\mathbb{Z}\to\mathbb{H}$ be the unique up to sign topological group isomorphism, let $\alpha\in\mathbb{R}/\mathbb{Z}$ be irrational, and define

$$\chi(k)=\gamma(\alpha k),\qquad A:=\chi(\mathbb{Z}).$$

Generic smooth expansion conjecture. There is a comeager subset $\Lambda\subseteq C^\infty(I)$, possibly depending on $\alpha$, such that for every $f\in\Lambda$:

1. $(\mathbb{R},+,\times,f)$ is o-minimal;
2. if $f\ne g$ are in $\Lambda$, then $(\mathbb{R},+,\times,f)$ and $(\mathbb{R},+,\times,g)$ are not interdefinable;
3. every $(\mathbb{R},+,\times,f)$-definable group is definably isomorphic to a semialgebraic group, and every definable homomorphism between semialgebraic groups is semialgebraic;
4. $(\mathbb{R},+,\times,A)$ is NIP and $\operatorname{Th}(\mathbb{R},+,\times)$ is an open core of $\operatorname{Th}(\mathbb{R},+,\times,A)$;
5. every $(\mathbb{R},+,\times,A)$-definable subset of $A^k$ is a finite union of sets $b\oplus n(X\cap A^k)$, with $X$ semialgebraic and $b\in A^k$.

In particular, the structure induced on $A$ by $(\mathbb{R},+,\times)$ is a dp-minimal expansion of $(\mathbb{Z},+,C_\alpha)$.