Arithmetic characterization of algebraic isomonodromy leaves

Establish that for a smooth projective family \mathscr{X}\to S over a finitely generated integral \mathbb{Z}-algebra R and a flat bundle (\mathscr{E},\nabla) on a fiber X/R, the leaf of the isomonodromy foliation through [(X,\mathscr{E},\nabla)] is algebraic if and only if it is integral, and that this holds if and only if the leaf is p-integral to order \omega(p) for almost all primes p, where \omega(p)/p\to\infty.

Background

This conjecture is a non-linear analogue of the Grothendieck–Katz p-curvature conjecture. It proposes arithmetic integrality conditions that precisely detect algebraicity of leaves of the isomonodromy foliation on the non-abelian de Rham moduli over a base S. It is intended to classify finite orbits under monodromy (π1(S)) via integrality and p-integrality conditions.