Katz's intersection-cohomology Newton–Hodge inequality

Let $X$ be a projective variety over a number field $F$, let $n$ be an integer, and let $F^s$ be a separable closure of $F$. For every prime number $\ell$ and maximal ideal $\mathfrak{p}$ of $\mathscr{O}_F$ of residue characteristic different from $\ell$, consider the Newton polygon $\operatorname{NP}(\operatorname{Frob}_{\mathfrak{p}}|_{IH^n(X\otimes_FF^s,\mathbb{Q}_{\ell})})$ and the Hodge–Tate polygon $\operatorname{HTP}(IH^n(X\otimes_FF^s,\mathbb{Q}_{\ell}))$.

Katz's intersection-cohomology Newton–Hodge inequality. There exists a finite set $S=S(X,n)$ of maximal ideals of $\mathscr{O}_F$ such that for every prime number $\ell$ and every maximal ideal $\mathfrak{p}$ outside $S$ with residue characteristic different from $\ell$,

$$\operatorname{NP}(\operatorname{Frob}_{\mathfrak{p}}|_{IH^n(X\otimes_FF^s,\mathbb{Q}_{\ell})})\ge \operatorname{HTP}(IH^n(X\otimes_FF^s,\mathbb{Q}_{\ell})).$$

The source notes that this is known when $X$ is smooth, by results of Katz, Messing, Mazur, and Faltings; the singular case is the conjectural part.