Basis conjecture for the quotient of the Kirchhoff Lie algebra

Let $\mathcal L_n$ be the Lie algebra generated by the Kirchhoff differences $\varkappa_{ij}$, and let $K_n\subset \mathcal L_n$ be the kernel of its action in the permutation representation of $S_n$. For indices $i_s$ satisfying $s+1\leq i_s\leq n$ for every $s=1,\dots,n-1$, define the repeated commutators

$$[\dots[[\varkappa_{1i_1},\varkappa_{2i_2}],\varkappa_{3i_3}],\dots,\varkappa_{n-1,i_{n-1}}].$$

Basis conjecture. One has

$$\dim \mathcal L_n/K_n=(n-1)!.$$

Moreover, the repeated commutators above, as the indices range over all such choices, form a basis of $\mathcal L_n/K_n$. This is posed as a structural problem about the Lie algebra and its symmetric-group representation; the supplied text gives no resolution status.