Generation conjecture for the Lie algebra of Kirchhoff differences

Let $\mathcal L_n$ be the Lie algebra generated by the Kirchhoff differences $\varkappa_{ij}$ in the group algebra $\mathbb C[S_n]$, with $1\leq i<j\leq n$. Generation conjecture. The Lie algebra $\mathcal L_n$ is generated by the Kirchhoff differences $\varkappa_{ij}$, $1\leq i<j\leq n$. This is presented as a conjectural structural description of the Lie algebra; the supplied text does not state whether it has been resolved.