Periodic Higgs conjecture for motivic parabolic Higgs bundles

Let $C$ be a smooth projective curve, and let $\mathrm{MHG}(C)$ and $\mathrm{PHG}(C)$ denote respectively the motivic and periodic parabolic Higgs categories from the source. Let

$$\iota:\mathrm{MHG}(C)\to\mathrm{PHG}(C)$$

be the inclusion functor. Periodic Higgs conjecture. The inclusion functor $\iota$ is essentially surjective, hence an equivalence of categories. Since objects of $\mathrm{MHG}(C)$ are semisimple, the source notes that this conjecture would imply the polystability conjecture and the periodic de Rham conjecture.