Conjecture F on algebraic integrability of foliations

Let $X$ be a normal complex quasi-projective variety, and let $\mathcal{F}\subset T_X$ be a coherent subsheaf closed under the Lie bracket. The distribution $\mathcal{F}$ is algebraically integrable if there is a dense open subvariety $U\subset X$ and a smooth morphism $f:U\to V$ such that $\ker df=\mathcal{F}|_U$. Choose a finitely generated $\mathbb{Z}$-subalgebra $R\subset\mathbb{C}$ over which $X$ and $\mathcal{F}$ have models; reduction modulo almost all primes means reduction modulo all maximal ideals of $R$ outside a proper closed subscheme of $\operatorname{Spec}R$. In characteristic $p$, a coherent Lie subalgebra is $p$-integrable if it is closed under taking $p$-th powers. Conjecture F. $\mathcal{F}$ is algebraically integrable if and only if its reduction is $p$-integrable modulo almost all primes $p$. This predicts that algebraic integrability of a foliation is exactly detected by its reductions in positive characteristic; the supplied source does not state whether the conjecture is resolved.