Langlands' conjecture with finite-field Laurent-series coefficients

Let $X$ be a smooth projective curve over a finite field $\BF_q$, let $\rho:\pi_1(X)\to GL_n(\bF)$ be as in de Jong's conjecture, and let $f_\rho$ denote a function on the set of rank-$n$ vector bundles on $X$. A cuspidal Hecke eigen-form with eigenvalues corresponding to $\rho$ satisfies

$$T_x^i(f_\rho)=\lambda_x^i f_\rho,$$

for every place $x$ and $i=1,\ldots,n$, with $\lambda_x^i=\operatorname{Tr}(\Lambda^i(\rho(Fr_x)))$.

Langlands' conjecture. Let $\rho$ be as in de Jong's conjecture. Then there exists a non-zero cuspidal Hecke eigen-form $f_\rho$ with eigenvalues corresponding to $\rho$.

The paper explains that de Jong's conjecture follows from this version of Langlands' conjecture; the analogous conjecture with $\mathbb{Q}_l$-coefficients is stated to be a theorem of Lafforgue.