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 $Z:\mathcal K\to\hat{\mathcal A}$, which takes value in the completion of space of chord diagrams mod 1T/4T relations, by a weight system(a function on $\hat{\mathcal A}$ that vanishes on $\bigoplus_{k\geq n}\mathcal A_k$ for some $n$), so the whole finite type invariants are equally powerful as $Z$, and we can show that $\hat Z:\hat{\mathcal K}\cong\hat{\mathcal A}$(because you can show that $Z:\mathcal K/\mathcal K_{n+1}\cong\bigoplus_{k\leq n}\mathcal A_k$), 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) Every invariant factors through a universal finite type invariant, say $\mathcal K\to\hat{\mathcal K}$.

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

(3)Every invariant $V$ can be wrtten as $V=\lim_{n\to \infty} V_n$ for $V_n$ type $n$.

(4)[Stone-Weierstrass]Every invariant $V$ can be approximated by finite type invariants pointwise, namely $V=\lim_{n\to \infty} V_n$ for finite type $V_n$(not necassarily type $n$)

(Rmk) I do think Vassiliev conjecture is equivalent to (1), and I don't think it implies (2),(3),(4), even if an invariant $V$ factors through $Z:\mathcal K\to \hat{\mathcal K}=\hat {\mathcal A}=\prod_n \mathcal{A}_n$, here is an annoying problem of convergence. I have no idea of wether (2),(3),(4) implies Vassiliev's conjecture or not.

I doubt (2),(3),(4) might be false: 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?