Katz's Lie-algebraic $p$-curvature conjecture

Let $X$ be a smooth connected variety over ${\bf C}$, let $V$ be a vector bundle on $X$ with integrable connection $\nabla$, and let $G_{\rm gal}$ be the Tannakian Galois group of $(V,\nabla)$ over the function field $K(X)$, with Lie algebra $\mathfrak g$. For almost every prime $\mathfrak p$ of a finitely generated model $R$, reduce $\mathfrak g$ and the $p$-curvature map $\psi_p(V/{\mathfrak p}V,\nabla)$ modulo $\mathfrak p$.

Katz's Lie-algebraic $p$-curvature conjecture. With this notation, $\mathfrak g$ is the smallest algebraic Lie subalgebra of $\operatorname{End}_{\mathcal O_X}V\otimes K(X)$ such that, for almost every prime $\mathfrak p$ of $R$, the reduction of $\mathfrak g$ modulo $\mathfrak p$ contains the image of $\psi_p(V/{\mathfrak p}V,\nabla)$.

This is presented as a more general conjecture describing the Lie algebra of the Tannakian Galois group through reductions of the $p$-curvature. The source gives no resolution status for this formulation.