Lando–Yang conjecture on primitive projections of chord diagrams

Let $D$ be a chord diagram, and let $u(D)$ be its projection to the subspace of primitives. Let $\tau(D)$ denote the length of the longest cycle in the intersection graph of $D$. A connected Jacobi diagram is a Jacobi diagram whose underlying graph is connected.

Lando–Yang conjecture. The projection $u(D)$ is a linear combination of connected Jacobi diagrams with at most $\tau(D)$ legs. In particular, for any Lie algebra $bfg$ and its associated weight system, the value on $u(D)$ has degree at most $\tau(D)$.

This conjecture generalizes an earlier conjecture of S. Lando for the $bfs l_2$ weight system, which bounds the degree by half the length of the largest cycle. The computations in the source provide evidence for the claim, including the case of chord diagrams whose intersection graph is $K_{2,n}$, but no resolution is given.