The knot complement conjecture for knots in closed oriented 3-manifolds

Let $Y$ be a closed oriented 3-manifold, and let $K_1$ and $K_2$ be knots in $Y$. Their exteriors are the manifolds obtained by deleting open tubular neighborhoods of the knots; they are orientation-preservingly homeomorphic when there is an orientation-preserving homeomorphism between them.

Knot complement conjecture. If $K_1$ and $K_2$ have orientation-preservingly homeomorphic exteriors, then there exists an orientation-preserving homeomorphism of $Y$ mapping $K_1$ to $K_2$.

This is the proposed extension of the Knot Complement Theorem from $S^3$ to arbitrary closed oriented 3-manifolds. The paper notes that it is known in some settings, including certain lens spaces and circle bundles over surfaces, but the general statement remains open.