Finite-monodromy conjecture for compatible rank 2 local systems

Let $X$ be a complete curve over $\mathbb{F}_{q}$, and suppose $\\{\mathscr{L}_{l}\\}_{l\ne p}$ is a $\mathbb{Q}$-compatible system of absolutely irreducible rank-$2$ local systems with trivial determinant.

Finite-monodromy conjecture. The compatible system $\\{\mathscr{L}_{l}\\}_{l\ne p}$ should have finite monodromy.

This conjecture predicts strong arithmetic rigidity for rational compatible systems on complete curves. The source does not state a resolution.