Semisimplicity conjecture for periodic parabolic Higgs and de Rham bundles

Let $C$ be a smooth projective algebraic curve over $\mathbb{C}$. A periodic parabolic Higgs/de Rham bundle means a periodic object of the corresponding parabolic Higgs or de Rham category. Semisimplicity conjecture. Any periodic parabolic Higgs/de Rham bundle over $C$ is semisimple. In particular, the source notes that a periodic parabolic Higgs bundle is conjectured to be polystable. The general assertion is not resolved in the supplied text.