I would guess that you would need to understand the transfinite lower central series of the mapping class group. A knot complement which could not be distinguished from a solid torus by any finite lower central series quotient of the mapping class group of a Heegaard surface would be n-trivial for all n, which if I'm not mistaken follows from work of [a subset of] Ted Stanford, Habiro, and Garoufalidis-Goussarov-Polyak. I don't see offhand why such an example could not exist.
Therefore, I think this question is open, and I can't see why it would be easy.

I would guess that you would need to understand the transfinite lower central series of the mapping class group. A knot complement which could not be distinguished from a solid torus by any finite lower central series quotient of the mapping class group Torelli group of a Heegaard surface would be n-trivial for all n, which if I'm not mistaken follows from work of [a subset of] Ted Stanford, Habiro, and Garoufalidis-Goussarov-Polyak. I don't see offhand why such an example could not exist.
Therefore, I think this question is open, and I can't see why it would be easy.