Arithmetic Simpson correspondence for periodic parabolic bundles

Let $C$ be a smooth projective curve. The grading functor $\operatorname{Gr}$ sends periodic parabolic de Rham bundles over $C$ to periodic parabolic Higgs bundles over $C$. Arithmetic Simpson correspondence. The functor

$$\operatorname{Gr}:\{mph{periodic parabolic de Rham bundles over }C\}\to\{\emph{periodic parabolic Higgs bundles over }C\}$$

is an equivalence of semisimple categories over $\mathbb{C}$. This is presented as a weaker version of the periodic Higgs conjecture.