The weight-system coefficient conjecture for ${\mathfrak{sl}}_2$

Let ${\cal M}$ be the graded commutative cocommutative Hopf algebra of chord diagrams, let ${\cal PM}_{2k}$ be its primitive subspace in degree $2k$, and let $\pi_{2k}:{\cal M}_{2k}\to{\cal PM}_{2k}$ be the projection along decomposable chord diagrams. Let $b^{{\mathfrak{sl}}_2}$ be the ${\mathfrak{sl}}_2$ weight system, let $c$ denote the quadratic Casimir parameter, and let $R_k$ be the weight system defined in the paper. For a chord diagram $d$ with $2k$ chords, write $[c^k]b^{{\mathfrak{sl}}_2}(\pi_{2k}(d))$ for the coefficient of $c^k$ in this polynomial.

Weight-system coefficient conjecture. For any chord diagram $d$ with $2k$ chords,

$$[c^k]b^{{\mathfrak{sl}}_2}(\pi_{2k}(d))=2R_k(d).$$

The preceding proposition implies that the relevant value is a polynomial of degree at most $k$, so the coefficient is well-defined. The conjecture relates the weight systems $R_k$ and $b^{{\mathfrak{sl}}_2}$; no resolution is supplied here.