The universal finite-type invariant conjecture for the embedding approximation

Let ${\mathbb I}=[0,1]$, let ${\rm Emb}({\mathbb I},{\mathbb I}^3)$ be the space of proper embeddings of the interval with fixed endpoint data, and let $C_n\colon {\rm Emb}({\mathbb I},{\mathbb I}^3)\to AM_n({\mathbb I}^3)$ be the approximating map. Write $\pi_0(C_n)$ for the induced map on connected components.

Universal finite-type invariant conjecture. The map

$$\pi_0(C_n)\colon\pi_0({\rm Emb}({\mathbb I},{\mathbb I}^3))\to \pi_0(AM_n({\mathbb I}^3))$$

is a universal additive type $n-1$ invariant over $\mathbb Z$.

The connected components of the embedding space are knot types, so this asserts that the approximation detects finite-type information universally at order $n-1$. The source presents this as a conjecture for three-manifolds; no resolution is supplied.