Let $f,g:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be smooth injective and let $n\leq m$. Let $k \in \mathbb{N}$, and let $\iota_m^{m+k}:\mathbb{R}^m\rightarrow \mathbb{R}^{m+k}$ be the canonical inclusion. Suppose also that $f(\mathbb{R}^n)\cong \mathbb{R}^n\cong g(\mathbb{R}^n)$ via some $C^{\infty}$-diffeomorphism.

Fix a compact subset $K\subseteq \mathbb{R}^n$. For what values of $k$, does there exist a homeomorphism $\phi:\mathbb{R}^{m+k}\rightarrow \mathbb{R}^{m+k}$ satisfying
$$
\iota_m^{m+k}\circ f(x)= \phi\circ \iota_m^{m+k}\circ g(x) \qquad (\forall x \in K)?
$$
It isn't difficult to see that $k\leq m+n$. However, what is the *smallest* such value of $k$ for which this holds? My intuition says 1...

*Reduction to Extension Problem*

I guess since $f$ and $g$ are homeomorphisms onto their image then, $h:=g\circ f^{-1}:f(K)\rightarrow g(K)$ is a homomorphism. So the problem reduces to finding an extension of $h$ to all of $\mathbb{R}^{m+k}$. But when does such an extension exist?