The Rozansky--Witten relation for generalized Fujiki constants in dimension at least eight

Let $X$ be a hyperkähler manifold of dimension $2n\geq 8$. For a characteristic class $\alpha$, write $C(\alpha)$ for its generalized Fujiki constant.

Rozansky--Witten generalized Fujiki relation.

$$\frac{C(\operatorname{ch}_4^2+120\operatorname{ch}_8)\,C(1)}{C(\operatorname{ch}_4)^2}=\frac{(5n+7)(2n-1)(2n-3)}{5(n+1)(2n-5)(2n-7)}.$$

The relation is motivated by requiring the Riemann--Roch polynomial to have real roots in arithmetic progression; the supplied text gives no resolution status.