Suppose $M$ is a closed 3-manifold and $K_1,K_2$ are two homotopic knots in $M$. That is, they are two embeddings $f_1,f_2\colon S^1 \to M$ such that there exists a homotopy $h\colon S^1 \times [0,1] \to M$ with $h(x,0) = f_1(x)$ and $h(x,1) = f_2(x)$. Suppose also that there is an orientation preserving homeomorphism $M - K_1 \to M - K_2$. Must $K_1$ and $K_2$ be isotopic in $M$?
A positive result seems like it would have fairly large implications, not to mention being necessarily more difficult than Gordon and Luecke's result (and is therefore unreasonable to expect here) so I am mainly wondering if there are any known or simple counterexamples.