The motivic control conjecture for hyper-Kähler varieties

Let $X$ be a hyper-Kähler variety and let $\mathrm{KS}(X)$ be its Kuga–Satake variety. Let $X$ and $Y$ be deformation-equivalent hyper-Kähler varieties, and suppose there is a Hodge isometry

$$H^2(X,\mathbb{Q})\xrightarrow{\sim}H^2(Y,\mathbb{Q}).$$

Motivic control conjecture. The variety $X$ should be motivated by $\mathrm{KS}(X)$, and the motives of $X$ and $Y$ should be isomorphic. These assertions express the expectation that the motive of a hyper-Kähler variety is controlled by its second cohomology. The paper proves the second assertion for homological motives of OG6-resolutions, while the general conjecture remains open.