Vanishing range for torsion in lower-central-series quotients

Let $A_n(\mathbb Z)$ be the free associative algebra on $n$ generators, and let $B_\ell(A_n(\mathbb Z))[m_1,\ldots,m_n]$ denote the multihomogeneous component of multidegree $(m_1,\ldots,m_n)$. Torsion vanishing-range conjecture. This component contains no torsion unless

$$\ell\le m_1+\cdots+m_n-3.$$

The claim predicts a degree bound below which torsion cannot occur in any lower-central-series quotient.