The one-step integration conjecture for $T4T$ and $TFI$ invariants

Let an invariant of chord diagrams satisfy the $T4T$ and $TFI$ relations. An invariant is integrated one step when it is lifted to an invariant at the next diagrammatic level. The one-step integration conjecture. Any invariant satisfying the $T4T$ and $TFI$ relations can be integrated one step to an invariant that satisfies the same relations. This is presented as the key step in the Hutchings combinatorial-topological approach to the Fundamental Theorem of Vassiliev invariants; the source does not establish it.