André–Baldassarri equivalence conjecture for arithmetic flat bundles

Let $X$ be a smooth algebraic variety over $\overline{\mathbb{Q}}$ and let $\mathscr{F}$ be a flat bundle on $X$. A $G$-connection is a flat bundle satisfying the arithmetic condition denoted by type $G$; globally nilpotent and almost everywhere nilpotent refer to the corresponding nilpotence conditions on $\mathscr{F}$. André–Baldassarri's equivalence conjecture. The following three conditions are equivalent to each other:

• $\mathscr{F}$ is a $G$-connection;
• $\mathscr{F}$ is globally nilpotent;
• $\mathscr{F}$ is almost everywhere nilpotent.

The conjecture relates three arithmetic classes of flat bundles. In the paper, the authors prove the equivalence for rigid flat bundles, giving an affirmative answer in that case; the general statement is therefore solved as stated in the source's context.