Positivity of signed Chern character numbers for irreducible hyper-Kähler manifolds

Let $M$ be a compact almost-complex manifold of real dimension $2n$, with Chern roots $x_1,\ldots,x_n$. Define its Chern character components by

$$\operatorname{ch}_i(M):=\frac{x_1^i+\cdots+x_n^i}{i!}\in H^{2i}(M;\mathbb{Q}).$$

For an integer partition $\lambda=(\lambda_1,\ldots,\lambda_{l(\lambda)})$ of weight $n$, define the Chern character number

$$\operatorname{Ch}_{\lambda}[M]:=\int_M\prod_{i=1}^{l(\lambda)}\operatorname{ch}_i(M)\in\mathbb{Q}.$$

Chern-character positivity conjecture. All signed Chern character numbers $(-1)^n\operatorname{Ch}_{2\lambda}$, where $|\lambda|=n$, of irreducible hyper-Kähler manifolds are positive. Positivity of these numbers is known for generalized Kummer varieties, but the assertion for all irreducible hyper-Kähler manifolds remains open.