Conjecture 3 on invariant 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 $\mathfrak{sl}_2J(D)$ be the associated free Lie algebra in the category with an $\mathfrak{sl}_2$-action by derivations. Let $H_k$ denote Lie algebra homology.

Conjecture 3. For every $k\geq 1$,

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

This is explicitly described as a weaker and more tractable version of Conjecture 2, yet the paper proves that it would imply Conjecture 1. It is established only in low degrees in the supplied discussion, so the general statement remains open.