Homological stability for automorphism groups of direct products

Let $H$ and $G$ be finitely generated groups, and let $F:U\! f\mathcal{G}\to \mathbb{Z}\operatorname{-Mod}$ be a coefficient system of finite degree. For each $n$, consider the homology group $H_i(\operatorname{Aut}(H\times G^n);F(H\times G^n))$. Direct-product stability conjecture. The map

$$H_i(\operatorname{Aut}(H\times G^n);F(H\times G^n))\longrightarrow H_i(\operatorname{Aut}(H\times G^{n+1});F(H\times G^{n+1}))$$

is an isomorphism for $n$ large. This extends known stability results for several classes of groups, including products of centre-free groups, free abelian groups, finite abelian groups, and right-angled Artin groups, while the general case remains open.