The mod-$p$ Artin–Tate integrality conjecture

Let $F$ be a totally real number field, and let $\rho: \operatorname{Gal}(\bar{\mathbb Q}/F) \rightarrow \operatorname{GL}_n(\bar{\mathbb Q}_p)$ be an irreducible representation cutting out a finite CM extension $E$ of $F$ that is not totally real. The value $L(0,\rho)$ is algebraic and nonzero. Let $\omega$ denote the action of $\operatorname{Gal}(\bar{\mathbb Q}/F)$ on the $p$-th roots of unity, and suppose that the mod-$\mathfrak p$ representation $\omega\otimes\bar\rho$ does not contain the trivial representation. Mod-$p$ Artin–Tate conjecture. Then

$$L(0,\rho)\in\bar{\mathbb Z}_p.$$

This is proposed as a mod-$p$ analogue of the expectation that an Artin $L$-function has a pole at $s=1$ precisely when its representation contains the trivial representation; it predicts the relevant $p$-integrality of special values and is open in the stated generality.