Exact-sequence conjecture relating homology-cylinder filtrations

Let $H$ be the first homology group used to define the tree and homology-cylinder filtrations, and let $\mathcal G_k^Y(\mathcal H_g)$ and $\mathcal G_k^w(\mathcal H_g)$ denote their degree-$k$ graded groups. For $k>1$, write $k=2l$ or $k=2l-1$ as appropriate. Exact-sequence conjecture. For $l>1$, the relationship between these graded groups should be given by the exact sequences

$$0\longrightarrow \mathcal G_{2l}^Y(\mathcal H_g)\longrightarrow \mathcal G_{2l}^w(\mathcal H_g)\longrightarrow L_{l+1}(H)/2L_{l+1}(H)\longrightarrow 0$$

and

$$H\otimes L_l(H)/2L_l(H)\longrightarrow \mathcal G_{2l-1}^Y(\mathcal H_g)\longrightarrow \mathcal G_{2l-1}^w(\mathcal H_g)\longrightarrow 0.$$

These sequences are proposed as the precise comparison between the two filtrations, incorporating the preceding conjectures. The source provides related rational and torsion exact sequences, but does not establish these sharper integral sequences.