Conjecture 2 on homology of the free Tits Lie algebra

Let $K$ be a field of characteristic zero, let $J(D)$ be the free Jordan algebra on $D\geq 1$ generators, and let $\mathfrak{sl}_2J(D)$ denote the corresponding free Lie algebra in the category of Lie algebras with an $\mathfrak{sl}_2$-action by derivations. Write $H_k$ for Lie algebra homology, and use superscripts $\mathfrak{sl}_2$ and $ad$ for the trivial and adjoint $\mathfrak{sl}_2$-isotypical components.

Conjecture 2. For every $k\geq 1$,

$$H_k(\mathfrak{sl}_2J(D))^{\mathfrak{sl}_2}=0,\qquad H_k(\mathfrak{sl}_2J(D))^{ad}=0.$$

The conjecture is motivated by the freeness of the Tits–Allison–Gao Lie algebra in the relevant category. The trivial-component vanishing is presented as a weaker, more tractable conjecture, while the full assertion remains unproved; the trivial component is known for $k=1,2,3$.