Is every $C^k$ embedding the restriction of a diffeomorphism on a subspace?

Question

Let $D$ be an open subset of $\mathbb{R}^d$ and $\phi:D\rightarrow \mathbb{R}^n$ a $C^k$ embedding. Is $\phi$ the restriction of some diffeomorphism (on $\mathbb{R}^n$) on the subspace $\mathbb{R}^d \times \{(0,\dots,0)\}$?

Two remarks that are consequences of the inverse function theorem:

1. The statement is locally true.

2. For $d=n$ the statement is true.