The Kuga--Satake Hodge conjecture

Let $V=H^{2r}(X,\mathbb{Q})$ be a Hodge structure of K3 type for some smooth projective variety $X$. Let $KS(V)$ be the Kuga--Satake variety associated to $V$, with an embedding of Hodge structures

$$\iota_V\colon V\hookrightarrow H^2(KS(V)\times KS(V),\mathbb{Q}).$$

Kuga--Satake Hodge conjecture. The embedding $\iota_V$ is induced by a correspondence in $A^\ast(X\times KS(V)\times KS(V))$. This is a special case of the Hodge conjecture; the paper proves it for the K3 surfaces under consideration, while the general assertion is not established here.