Hodge-theoretic Hopf conjecture for compact Kähler manifolds with nef tangent bundle

Let $X$ be a compact Kähler manifold of complex dimension $n$ with nef tangent bundle $TX$, and define

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

Hodge-theoretic Hopf conjecture. For every $p$ with $0\leq p\leq n$,

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

The claim is a Hodge-theoretic analogue of the Kähler consequence of Hopf's curvature conjecture. It holds for compact Kähler manifolds with non-negative bisectional curvature, and the general nef-tangent-bundle case remains open.