It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle?

Denote by $S$ the space $\Bbb R^2\setminus\{0\}$ (the plane minus the origin) under the induced topology. Suppose $C$ is a loop that is smoothly homotopic in $S$ to the unit circle. Map $C$ to $\Bbb R^2$ using the obvious inclusion map. By abuse of notation we will also denote this loop in $\Bbb R^2$ as $C$.

If an additive subgroup of $\Bbb R^2$ contains $C$, is it necessarily all of $\Bbb R^2$?