The mod-$p$ Artin $L$-value reduction conjecture

Let $F$ be a totally real number field, and let $\bar\rho: \operatorname{Gal}(\bar{\mathbb Q}/F)\rightarrow\operatorname{GL}_n(\bar{\mathbb F}_p)$ be a semisimple modular representation cutting out a finite CM extension $E$ of $F$ that is not totally real. Suppose that $\omega\otimes\bar\rho$ does not contain the trivial representation, where $\omega$ is the action on the $p$-th roots of unity. Mod-$p$ Artin $L$-value conjecture. One can define $\bar L(0,\bar\rho)\in\bar{\mathbb F}_p$ such that

$$\bar L(0,\bar\rho_1+\bar\rho_2)=\bar L(0,\bar\rho_1)\bar L(0,\bar\rho_2)$$

for any two such representations, and such that whenever $\bar\rho$ is the semisimplified mod-$p$ reduction of a representation $\rho$ as in the mod-$p$ Artin–Tate conjecture, the reduction of $L(0,\rho)\in\bar{\mathbb Z}_p$ equals $\bar L(0,\bar\rho)$. This conjecture seeks a well-defined multiplicative mod-$p$ special value compatible with characteristic-zero Artin $L$-values; its status is open.