General affine group is conjugacy-closed in self-diffeomorphism group

Statement
Let $$GA(n,\R)$$ denote the general affine group: the group ofall affine transformations of $$\R^n$$, i.e. all transformations of the form:

$$x \mapsto Ax + b$$

where $$A$$ is a $$n \times n$$ invertible matrix over $$\R$$ and $$b$$ is a vector.

Let $$Diff(\R^n)$$ denote the self-diffeomorphism group of $$\R^n$$. Then, any two elements of $$GA(n,\R)$$, which are conjugate in $$Diff(\R^n)$$, are conjugate in $$GA(n,\R)$$.

In fact the above result is true even if we place $$Diff(\R^n)$$ by the group of self-maps which are $$C^1$$ and have a $$C^1$$ inverse.

Related facts
It can also be shown that the general affine group is not conjugacy-closed in self-homeomorphism group. Since the property of being conjugacy-closed is a transitive subgroup property, this shows that the self-diffeomorphism group of $$\R^n$$, and in fact even the group of $$C^1$$-maps of $$\R^n$$ is not conjugacy-closed in the self-homeomorphism group.