de Jong's conjecture on finite geometric monodromy in characteristic $p$

Let $k$ be a finite field of characteristic $\ell\ne p$, let $X$ be a geometrically connected smooth curve over $k$, and let

$$1\to \pi_1(\overline{X})\to \pi_1(X)\to G_k\to 1$$

be the exact sequence for the arithmetic fundamental group, where $\overline{X}$ is the base change to an algebraic closure of $k$. Let $F$ be a local field of characteristic $p$. de Jong's conjecture. Every continuous representation

$$\rho:\pi_1(X)\to \mathrm{GL}_n(F)$$

has finite geometric monodromy: $\rho(\pi_1(\overline{X}))$ is finite. The source notes that this conjecture is used in the proof and records its solution by Gaitsgory for $p>2$ and by de Jong for $n\le 2$.