The semisimple monodromy conjecture for vanishing p-curvature

Let $X$ be a smooth variety over $\mathbb{C}$ and let $(V,\nabla)$ be a vector bundle with integrable connection on $X$. For almost all primes $p$, reduce the data to characteristic $p$ and consider its $p$-curvature.

Semisimple monodromy conjecture. If almost all $p$-curvatures of $(V,\nabla)$ vanish, then $(V,\nabla)$ has semisimple monodromy.

This is an isomonodromy statement implied by the p-curvature conjecture: algebraic solutions should force the associated monodromy representation to be semisimple. The supplied context does not state that this conjecture has been proved separately.