The periodic de Rham conjecture for irreducible bundles over curves

Let $U$ be a smooth complex curve. A periodic de Rham bundle on $U$ is a de Rham bundle whose spreading-out has reductions that are periodic in positive characteristic. A de Rham bundle is motivic if it arises from the motivic construction considered in the paper.

Periodic de Rham conjecture. Any irreducible periodic de Rham bundle over $U$ is motivic.

The paper proves the converse implication, namely that motivic de Rham bundles are periodic. This conjecture asserts the remaining direction and is presented as the main question for hyperbolic curves.