A finiteness theorem for mod $p$ Galois representations over global function fields

Let $p$ be an odd prime number and let ${\bar{\mathbb{F}}}_p$ be a fixed algebraic closure of the finite field of order $p$. Let $K$ be a global function field of characteristic different from $p$ and let $G_K$ be the absolute Galois group of $K$. We prove that there are only finitely many isomorphism classes of continuous geometric semisimple representations $\rho :G_K\to {\mathrm{G}\mathrm{L}}_n({\bar{\mathbb{F}}}_p)$ such that their Artin conductors are bounded. It is worth emphasizing that we do not need to assume that $p$ does not divide $n$.