The canonicality conjecture for the weight systems $W_n$

Let $W_n$ be the weight system defined in the paper, let $Z$ denote the Kontsevich integral, and let $\nabla_n$ denote the degree-$n$ coefficient of the multivariable Conway potential function. Canonicality conjecture. $W_n$ is canonical for every $n$, meaning

$$W_n\circ Z=\nabla_n.$$

The case $n=1$ is known: as pointed out by D. Bar-Natan and M. Garoufalidis, $W_1\circ Z$ coincides with $\nabla_1$. The conjecture asserts that this compatibility holds for all $n$.