Beauville's conjecture on polynomial relations among divisors on hyper-Kähler manifolds

Let $M$ be a projective hyper-Kähler manifold, and let $d_1,\dots,d_r$ be divisor classes on $M$. A polynomial cohomological relation

$$P(d_1,\ldots,d_r)=0$$

in $H^*(M,\mathbb{Q})$ is also a relation in ${\rm CH}(M)$.

Beauville's conjecture. Any polynomial cohomological relation among divisor classes on $M$ that holds in $H^*(M,\mathbb{Q})$ already holds in ${\rm CH}(M)$.

The conjecture predicts that the subring generated by divisor classes has the same relations in the Chow ring as in cohomology. In the appendix, it is discussed as a consequence of the paper's results on LSV varieties; the supplied text does not establish its general resolution.