Affine action conjecture for groups acting on the line with finitely many fixed points

Let $N\in \mathbb{N}$ and let $G\leq {\mathrm{Homeo}}(\mathbb{R})$ be a subgroup with at most $N$ fixed points. Let $I\subset \mathbb{R}$ be a maximal interval without global fixed points for $G_+$. Affine action conjecture. The restriction of the action of $G$ to $I$ is semi-conjugate to the action by affine transformations.

The affine transformation group $\mathrm{Aff}(\mathbb{R})$ is the only known example, for any $N\geq 1$, of a group with at most $N$ fixed points acting minimally on the real line. The conjecture proposes that affine actions account for all such actions on maximal intervals without global fixed points.