Algebraicity of vector fields in characteristic zero and characteristic $p>0$

I will discuss a conjecture of Shepherd-Baron and Ekhedal (unpublished) that a distribution $E$ on a variety over $\mathbb{Q}$ is algebraic (i.e. most leaves are algebraic) if (and only if) for almost all primes $p$, the reduction $E_p$ is algebraic. This is a joint work in progress with D. Leshchiner