The nullhomotopic knot complement conjecture

Let $Y$ be a closed oriented 3-manifold, and let $K_1$ and $K_2$ be nullhomotopic knots, meaning knots representing the trivial element of $\pi_1(Y)$. Their exteriors are obtained by deleting open tubular neighborhoods of the knots; they are orientation-preservingly homeomorphic when there is an orientation-preserving homeomorphism between them.

Nullhomotopic 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 $K_2$.

The paper states that Boileau's nontrivial surgery conjecture implies this nullhomotopic knot complement problem. It is therefore a consequence of the broader surgery question and is not resolved in the generality stated here.