André–Christol conjecture for formal flat sections

Let $R\subset\mathbb{C}$ be a finitely generated $\mathbb{Z}$-algebra, let $X$ be a smooth $R$-scheme, let $(\mathscr{E},\nabla)$ be a flat vector bundle on $X/R$, let $x\in X(\mathbb{C})$, and fix $v\in\mathscr{E}_x$. André–Christol conjecture for flat sections. The formal flat section of $\mathscr{E}$ through $v$ is algebraic if and only if it is integral, if and only if it is $\omega(p)$-integral. This is the linear specialization of the main conjecture and is described as largely open, despite verification in numerous geometric examples.