The extension conjecture for periodic de Rham bundles

Let $U=C-D$ be a smooth complex curve, and let $(V,\nabla,Fil)$ be an irreducible periodic de Rham bundle on $U$. Its canonical parabolic extension is an adjusted parabolic de Rham bundle on the log curve $C_{\mathrm{log}}$.

Extension conjecture. The canonical parabolic extension of an irreducible periodic de Rham bundle over $U$ is again periodic.

Periodicity implies global nilpotence, and Katz's monodromy theorem gives regular singularities and rational residues, allowing the canonical parabolic extension to be defined. The conjecture further predicts that a periodic Hodge filtration is, up to an index shift, induced by the Simpson filtration on this extension.