Khare–Moon finiteness conjecture for mod $p$ Galois representations over number fields

Let $K$ be a number field, let $n$ be a positive integer, let $\mathfrak{R}$ be a nonzero ideal of the ring of integers of $K$, and let $G_K$ be the absolute Galois group of $K$. For a continuous semisimple representation

$$\rho:G_K\to \mathrm{GL}_n(\overline{\mathbb{F}}_p),$$

write $\mathfrak{R}(\rho)$ for its prime-to-$p$ Artin conductor. Khare–Moon finiteness conjecture. There are only finitely many isomorphism classes of such representations for which $\mathfrak{R}(\rho)$ is bounded by $\mathfrak{R}$. This conjecture is a finiteness statement for mod $p$ Galois representations with bounded ramification; the source records results in several special cases and proves it when $p$ is odd or $n=2$.