I have a statement which is related to the smooth Jordan-Schoenflies Theorem. I can verify it in some simple cases.
The setup is: Let there be a $C^1$ immersion $\gamma : S^1 \to \mathbb{R}^2$. Assume it is in general position, i.e. there are a finite number of self-intersections and all of them are transverse double-points. Then the complement $\mathbb{R}^2 - \gamma(S^1)$ has a finite number of open connected components and the winding number of $\gamma$ is constant on each component. Let the components be $U_i$ for $i=1,\ldots,N$ and the corresponding winding numbers be $w_i$.
The statement is:
$\gamma$ has a $C^1$ extension $\Gamma : D^2 \to \mathbb{R}^2$ so that if $y \in U_i$, then $y$ is a regular value of $\Gamma$, and the signed sum over the points in $\Gamma^{-1}(y)$ is equal to $w_i$. The signs are the usual ones: +1 if $\Gamma$ preserves orientation at the point and -1 if $\Gamma$ reverses orientation.
Is it true ?
In the case that $\gamma$ is a simple closed curve (with no self-intersections), it is true by the smooth Jordan-Schoenflies Theorem. There are only 2 components $U_1$ and $U_2$ (inside and outside), with winding numbers $w_1=\pm 1$ and $w_2=0$.
This might be related to the Reidemeister moves in knot theory.