This question question is moved from math stackexchange which I posted several days ago without an answer.

Background(ignore this paragraph if you know finite type invariants well): Recall that a finite type invariant of degree $n$ is an invariant $V$ such that $V^{(n+1)}=0$ where $V^{(n+1)}$ is the extension of $V$ to $(n+1)$-singular knots. Fix a base field $\mathbb F$ of characteristc $0$. Let $\mathcal V_n$ be the space of type $n$ invariants and $\mathcal K_n$ be the linear space spanned by the complete resolution of $n$-singular knots(these are commone zeros of type $n-1$ invariants), and let $\mathcal V$ and $\mathcal K$ be the corresponding filtered space of finite type invariants and knots. The relations not distinguished by the whole finite type invariants $\mathcal V=\bigcup\mathcal V_n$ is $\bigcap\mathcal K_n=\text{ker} (\mathcal K\to\hat{\mathcal K})$, and Vassiliev conjecture is that $\mathcal K\to\hat{\mathcal K}$ is injective. Note that every finite type invariant factors through the Kontsevich integral $Z$ by a weight system, so the whole finite type invariants are equally powerful as $Z$, and we can show that the completion of space of chord diagrams mod 1T/4T relations $\hat{\mathcal A}=\hat{\mathcal K}$(because you can show that $\mathcal K/\mathcal K_{n+1}=\bigoplus_{k\leq n}\mathcal A_k$), and Vassiliev conjecture is that $Z$ injective.

My question: What is the relation of Vassiliev's conjecture with the following conditions(at least when $\mathbb F=\mathbb R$ so that the limit is defined):

(1)[Taylor expansion] Every invariant $V$ has an expansion $V=\sum_n V_n$ of finite type invariants for $V_n$ type $n$.

(2)[Stone-Weierstrass] Every invariant $V$ can be approximated by finite type invariants, namely $V=\lim_{n\to \infty} V_n$ for $V_n$ type $n$.

(3) Every invariant factors through a universal finite type invariant, say $\mathcal K\to\hat{\mathcal K}$.

(Rmk) I don't think Vassiliev's conjecture implies (1) and (2), even if an invariant $V$ factors through $Z:\mathcal K\to \hat{\mathcal K}=\hat {\mathcal A}=\prod_n \mathcal{A}_n$, we cannot reduce a general linear functional on a product of spaces to a form $(x_j)_j\mapsto (a_jx_j)_j$, plus here is an annoying problem of convergence. I have no idea of wether (1),(2),(3) implies Vassiliev's conjecture or not.

Another question, are there known examples of an invariant that cannot expand as power series of finite type invariants or as a limit of finite type invariants?