Kuga-Satake Hodge conjecture for projective hyper-Kähler varieties

Let $X$ be a projective hyper-Kähler variety, and let $H^2_{\mathrm{tr}}(X)$ denote its transcendental second cohomology. Let $\mathrm{KS}(X)$ be the Kuga-Satake variety associated with $H^2_{\mathrm{tr}}(X)$ and the Beauville–Bogomolov form. An algebraic cycle $\zeta$ on

$$X\times \mathrm{KS}(X)\times \mathrm{KS}(X)$$

should exist such that the associated correspondence induces an embedding of Hodge structures

$$\zeta_*\colon H^2_{\mathrm{tr}}(X)\hookrightarrow H^2(\mathrm{KS}(X)\times \mathrm{KS}(X)).$$

This is the Hodge-theoretic algebraicity assertion underlying the Kuga–Satake correspondence: the natural embedding into the second cohomology of a product of Kuga–Satake varieties should be induced by an algebraic cycle. Its status is not resolved in the supplied source context.