Grothendieck's conjecture on algebraic solutions and reductions modulo primes

Let $\mathscr{L}\in\mathbb{Q}(x)\langle\partial_x\rangle$ be the differential operator attached to the paper's linear differential equation, and let $\mathscr{L}_p$ denote its reduction modulo a prime $p$. Say that $\mathscr{L}$ has a full basis of algebraic solutions when its solution space in $\overline{\mathbb{Q}(x)}$ has dimension equal to its order; similarly, say that $\mathscr{L}_p$ has a full basis of rational solutions when its solution space in $\mathbb{F}_p(x)$ has maximal dimension over the field of differential constants $\mathbb{F}_p(x^p)$. Grothendieck's conjecture. The following properties are equivalent: (1) $\mathscr{L}$ has a full basis of algebraic solutions; (2) for almost all primes $p$, $\mathscr{L}_p$ has a full basis of rational solutions. This is a central conjecture relating algebraic solutions in characteristic zero to arithmetic behavior after reduction modulo primes. The source notes that it has been proved for important classes of differential equations, while the general statement remains open.