Moon–Taguchi finiteness conjecture for geometric mod $p$ representations over function fields

Let $K$ be a global function field of characteristic different from $p$, with absolute Galois group $G_K$. Let $n$ be a positive integer and let $\mathfrak{R}$ be an effective divisor of $K$. A representation is geometric when the fixed field of the kernel contains no constant field extension of $K$. Moon–Taguchi finiteness conjecture. There are only finitely many isomorphism classes of continuous semisimple representations

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

whose Artin conductor $\mathfrak{R}(\rho)$ divides $\mathfrak{R}$ and which are geometric. The conjecture is the function-field analogue of the number-field finiteness conjecture; the source states that the paper proves it when $p$ is odd or $n=2$, removing the earlier restriction $p\nmid n$.