The Weil-generic vanishing conjecture for lower central series quotients

Let

$$A=\mathbb{C}\langle x,y\rangle/(P),$$

where $P$ is a noncommutative polynomial of degree $d$ whose abelianization is square-free, and let $B_m$ and $N_m$ denote the associated lower central series quotients with graded pieces $B_m[r]$ and $N_m[r]$. Say that $P$ is Weil generic if it lies outside a countable union of hypersurfaces in the space of noncommutative polynomials of degree $d$.

Weil-generic vanishing conjecture. For Weil generic $P$ of degree $d$,

$$B_m[r]=N_m[r]=0$$

for $r\geq 2d+m-3$. Equivalently, the width of the interval of nonzero values is at most $2d-3$.

The preceding proposition gives the weaker bound $r\geq 2d+2m-5$ for $B_m[r]=0$ and $r\geq 2d+2m-4$ for $N_m[r]=0$. Computational evidence suggests that the sharper bound holds, but the source does not report a proof.