Isomorphism conjecture for the tree-to-quasi-Lie map

Let $H$ be a free abelian group, let $L'(H)$ be the free quasi-Lie algebra on $H$, and let $\mathcal A_k^t(H)$ be the abelian group generated by labelled planar binary trees of the relevant degree modulo the anti-symmetry and IHX relations. Let $\mathsf D'_k(H)$ be the kernel of the bracket map

$$H\otimes L'_{k+1}(H)\longrightarrow L'_{k+2}(H).$$

The map $\eta'_k:\mathcal A_k^t(H)\to\mathsf D'_k(H)$ sends each labelled binary tree to the sum of the rooted trees obtained by designating each leaf as the root. Isomorphism conjecture. The map

$$\eta'_k:\mathcal A_k^t(H)\longrightarrow\mathsf D'_k(H)$$

is an isomorphism for every $k$.

The source proves that $\eta'_k$ is a split surjection and that its kernel is torsion annihilated by $k+2$, with a parity-dependent description. The conjecture asks that this torsion kernel always vanish; no resolution is supplied.