Polystability conjecture for periodic Higgs bundles

Let $C$ be a smooth projective complex curve, and let a periodic Higgs bundle on $C$ mean a Higgs bundle satisfying the source's periodicity condition. Polystability conjecture. Any periodic Higgs bundle over $C$ is polystable. The source relates this claim to the stronger semisimplicity conjecture; no resolution is given in the supplied text.