Periodic Higgs conjecture

Let $C$ and $D$ be as in the source's theorem on periodicity for motivic Higgs bundles, and let $C_{\mathrm{log}}$ denote the corresponding logarithmic curve. A periodic parabolic Higgs bundle is a parabolic Higgs bundle satisfying the source's periodicity condition, while a motivic parabolic Higgs bundle is one arising from the relevant parabolic Gauß–Manin construction. Periodic Higgs conjecture. Any periodic parabolic Higgs bundle over $C_{\mathrm{log}}$ is motivic. The conjecture is known for rank-one objects, but the general case remains open.