Hodge-theoretic Singer–Hopf conjecture for compact Kähler manifolds

Let $X$ be a compact Kähler manifold of complex dimension $n$, and write

$$\chi^p(X):=\chi(X,\Omega_X^p),$$

where $\Omega_X^p$ is the sheaf of holomorphic $p$-forms. Assume that $X$ is aspherical or has a nef cotangent bundle. Hodge-theoretic Singer–Hopf conjecture. For every integer $p$ with $0\leq p\leq n$,

$$(-1)^{n-p}\cdot\chi^p(X)\geq 0.$$

This enhances the Euler-characteristic sign conjecture through the individual holomorphic Euler characteristics. It is proved in several special cases, including relevant Kähler-hyperbolic, Kähler-nonelliptic, and low-dimensional nef-cotangent settings, but is not resolved in general.