The invariant-homology vanishing conjecture for free Allison-Benkart-Gao Lie algebras

Let $k$ be the ground field, let $D$ be a non-negative integer, and let $A(D)$ be the free alternative algebra with $D$ generators. The homology groups $H_r(\mathcal{ABG}(A(D)))$ carry an $sl_3(k)$-action, and $H_r(\mathcal{ABG}(A(D)))^{sl_3}$ denotes their invariant submodule.

Invariant-homology conjecture. As $sl_3(k)$-modules,

$$H_r(\mathcal{ABG}(A(D)))^{sl_3}=0$$

for any $r\geq1$.

This conjecture concerns only the invariant part of the homology and is stronger in its range of degrees than the low-degree vanishing currently established in the paper. Its general validity remains open.