Does somebody have a reference (or an argument why it should be true) for the following statement?

“Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 \to S^3$ which takes $K$ to $K'$, then there is also an ambient isotopy $\eta : (S^3,K) \to (S^3,K')$.”

It seems to me that this statement is *always* assumed when citing the Gordon–Luecke theorem from [Gordon, Luecke: Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989), 371-415].

Namely, the authors define knot equivalence by the existence of an $h$ as in the statement (actually they work with homeomorphisms $h$ that are not necessarily orientation-preserving, but they say that everything still works in the orientation-preserving case), while everybody else seems to use the ambient-isotopy definition, even when citing the Gordon–Luecke theorem.