Mixed-Hodge-module extension of the Hodge-theoretic Singer–Hopf conjecture

Let $X$ be a compact Kähler or complex projective manifold that is aspherical or has a nef cotangent bundle, and let $\mathcal{M}$ be a mixed Hodge module on $X$. For an integer $p$, let $Gr_F^pDR(\mathcal{M})$ denote the $p$-th graded piece of the de Rham complex associated to $\mathcal{M}$ with respect to the Hodge filtration. Mixed-Hodge-module Singer–Hopf conjecture. For every integer $p$,

$$\chi\bigl(X,Gr_F^pDR(\mathcal{M})\bigr)\geq 0.$$

The claim generalizes the preceding Hodge-theoretic inequality and is known for smooth subvarieties of, and more generally varieties admitting a finite morphism to, an abelian variety. It remains open in the stated generality.