The abelian motive conjecture for hyper-Kähler varieties

Let $X$ be a hyper-Kähler variety, and let $\mathfrak{h}(X)$ denote its motive. For each degree $i$ and prime $ell$, let $\mathbf{G}_{\ell,i}(X)$ be the $ell$-adic algebraic monodromy group and let $\operatorname{\mathbf{MT}}_i(X_{\mathds{C}})$ be the Mumford–Tate group of the Betti Hodge structure. The abelian motive conjecture. The motive $\mathfrak{h}(X)$ is an abelian motive. In particular, there is a canonical injective homomorphism

$$\mathbf{G}_{\ell,i}(X)^{\circ} \hookrightarrow \operatorname{\mathbf{MT}}_i(X_{\mathds{C}})_{\mathds{Q}_{\ell}}.$$

The conjecture expresses the expectation that the full motive of a hyper-Kähler variety is governed by abelian motives, extending the known abelian-motive result for its second-degree motive. It has been verified for the four known deformation types $\mathrm{K3}^{[n]}$, $\operatorname{Kum}_n$, $\operatorname{OG}6$, and $\operatorname{OG}10$.