Euler-operator injection conjecture for lower central series quotients

Let $B_k$ be multigraded, and write $B_k[d_1,\ldots,d_n,0,0]$ for its multidegree component. Let $E:=\sum x_i\partial_i$ be the Euler operator, and set

$$D:=[x_{n+1},x_{n+2}]E\in\widetilde{W_{n+2}}.$$

Euler-operator injection conjecture. The operator $D$ is an injection

$$D:B_k[d_1,\ldots,d_n,0,0]\hookrightarrow B_k[d_1,\ldots,d_n,1,1].$$

The conjecture is stated as verifiable for $\bar{B}_1$, $B_2$, and $B_3$ from their explicit descriptions, but remains open in general.