Beauville's divisor-generated cycle class conjecture for hyper-Kähler varieties

Let $X$ be a projective hyper-Kähler manifold, and let $\operatorname{CH}^*(X)$ denote its Chow ring with $\mathbb{Q}$-coefficients. Beauville's conjecture. The cycle class map is injective on the subalgebra of $\operatorname{CH}^*(X)$ generated by divisors. This is a concrete consequence of the expected Beauville splitting and would identify the divisor-generated Chow subalgebra with its image in cohomology.