Hard Lefschetz and Hodge–Riemann conjecture for products of Chern classes of Kähler vector bundles

Let $X$ be a compact Kähler manifold of dimension $n$, and let $E_1,\ldots,E_k$ be Kähler vector bundles on $X$, with $e_i=\operatorname{rank}(E_i)$. Let $p,q$ be nonnegative integers satisfying

$$p+q+e_1+\cdots+e_k=n.$$

Hard Lefschetz and Hodge–Riemann conjecture. The class

$$c_{e_1}(E_1)\cdots c_{e_k}(E_k)$$

satisfies the hard Lefschetz property and the Hodge–Riemann relation.

This conjecturally extends the hard Lefschetz and Hodge–Riemann results proved in the preceding theorem for one Kähler vector bundle together with additional Kähler classes. The claim concerns the interaction of top Chern classes of several Kähler vector bundles on a compact Kähler manifold.