The positivity conjecture for products of generalized Fujiki constants

Let $X$ be a hyperkähler manifold of dimension $2n$, and let $k_1,\dots,k_r\in\mathbf Z_{>0}$ with

$$k\coloneqq\sum_i k_i\leq n.$$

For a characteristic class $\alpha$, let $C(\alpha)$ denote its generalized Fujiki constant.

Generalized Fujiki positivity conjecture.

$$(-1)^kC(\operatorname{ch}_{2k_1}\cdots\operatorname{ch}_{2k_r})>0 \quad\text{and}\quad C(c_{2k_1}\cdots c_{2k_r})>0.$$

This conjecture generalizes the observed degree-four behavior and asks for positivity for arbitrary products of even Chern-character and Chern-class components; the question was also asked independently by Cao, Oberdieck, and Toda, and the supplied text gives no resolution status.