The periodic Higgs conjecture for parabolic Higgs bundles

Let $C$ be a smooth projective curve over $\mathbb{C}$, and let $D\subset C$ be a reduced effective divisor. A parabolic Higgs bundle over $(C,D)$ is called periodic when it is periodic under the relevant Higgs–de Rham flow. Periodic Higgs conjecture. Any stable periodic parabolic Higgs bundle over $(C,D)$ is motivic. This generalizes the periodic Higgs conjecture previously formulated for the setting considered in the paper; the rank-one case with empty $D$ is known, but the general statement remains open.