Given $f,g\colon\mathbb{R}^n\hookrightarrow\mathbb{R}^d$ which are smooth embeddings with the same image, does there exist a diffeomorphism $h \in \mathrm{Diff}(\mathbb{R}^n)$ such that $f=g\circ h$?
What I want is an equivalence relation between embeddings with the same image. If there is no such equivalence, so be it.
I did see this question, but I'm interested in precomposition.