W_n-module conjecture for consecutive quotients of the free associative Lie algebra

Let $A_n$ be the free associative algebra on $n$ generators, regarded as a Lie algebra, and let $A_{n,k}$ be its consecutive lower-central quotients. Let $W_n$ be the Lie algebra of polynomial vector fields on $\mathbb{C}^n$, let $\mathcal C$ be the class of $W_n$-modules introduced in the source, and set

$$gr^+(A_n)=\bigoplus_{k>1}A_{n,k}.$$

W_n-module conjecture. Every quotient $A_{n,k}$ for $k>1$ is a $W_n$-module in the class $\mathcal C$; the Lie bracket on $gr^+(A_n)$ is $W_n$-equivariant.