Injectivity conjecture for the free quasi-Lie algebra's 2-torsion map

Let $H$ be a free abelian group, and let $L(H)$ and $L'(H)$ denote the free Lie algebra and free quasi-Lie algebra on $H$, respectively. For $l\geq 1$, write $L_l(H)$ and $L'_{2l}(H)$ for their homogeneous components. The natural map

$$L_l(H)/2L_l(H)\longrightarrow L'_{2l}(H)$$

comes from the exact sequence relating $L'_{2l}(H)$ and $L_{2l}(H)$. Injectivity conjecture. The map

$$L_l(H)/2L_l(H)\longrightarrow L'_{2l}(H)$$

is a monomorphism for every $l$.

This would determine the kernel in the even-degree comparison between the free quasi-Lie and free Lie algebras. The source verifies the assertion for $l=1$ but gives no resolution for general $l$.