The geometric characterization of knots trivial to all finite orders

A knot $K$ is $n$-trivial if its Vassiliev invariants of orders at most $n$ vanish, and $n$-hyperbolic if it bounds an $n$-hyperbolic Seifert surface, meaning that the complement of the surface has the prescribed lower-central-series resemblance to the complement of a Seifert surface of a trivial knot. For every $n\in\mathbb{N}$, these notions are considered simultaneously.

Geometric characterization conjecture. A knot $K$ is $n$-trivial for all $n\in\mathbb{N}$ if and only if it is $n$-hyperbolic for all $n\in\mathbb{N}$.

The conjecture asks whether $n$-hyperbolicity gives a complete geometric characterization of knots whose Vassiliev invariants vanish in every finite order. The paper proves that $n$-hyperbolicity for all $n$ implies the vanishing of all Vassiliev invariants, while the converse is conjectural and is supported by further evidence cited by the authors.