Morita's conjecture on the abelianization of derivations of a free Lie algebra

Let $F_n$ be the free group of rank $n$, let $\mathcal{L}_n$ be the free Lie algebra associated with the lower central series of $F_n$, and let $\mathrm{Der}^+(\mathcal{L}_n)$ be the positive-degree derivation Lie algebra. Write $(\mathrm{Der}^+(\mathcal{L}_n))^{\mathrm{ab}}$ for its abelianization, $H$ for the abelianization of $F_n$, $H^*$ for its dual, and $S^kH$ and $\Lambda^2H$ for the symmetric and exterior powers. Morita's conjecture. For $n\geq3$,

$$(\mathrm{Der}^+(\mathcal{L}_n))^{\mathrm{ab}}\cong (H^*\otimes_{\mathbf{Z}}\Lambda^2H)\oplus\bigoplus_{k\geq2}S^kH.$$

This predicts that the abelianization is generated by the degree-one part together with Morita's trace classes; the analogous statement for the Chen Lie algebra is established in the source, while the free-Lie-algebra version remains open.