Grothendieck's p-curvature conjecture for algebraic connections

Let $X$ be a smooth quasi-projective variety over $\mathbb C$, and let $(E,\nabla)$ be an integrable algebraic connection. Choose a finite-type scheme $S$ over $\mathbb Z$ over which the connection descends to $(X_S,(E_S,\nabla_S))$ with good reduction at closed points $s\in S$. Grothendieck's p-curvature conjecture. The connection $(E,\nabla)$ has a full set of algebraic solutions if and only if there is a dense open $S^\circ\subset S$ such that, for every $s\in S^\circ$, the reduction $(E_s,\nabla_s)$ on $X_s$ has a full set of solutions, equivalently its $p$-curvature vanishes. This is the general formulation attributed to Grothendieck; the source does not state a resolution.