The weak periodic de Rham conjecture for parabolic bundles

Let $C_{\mathrm{log}}$ be a log curve. Let $\mathrm{PDR}^{ss}(C_{\mathrm{log}})$ be the category of completely reducible periodic parabolic de Rham bundles, and let $\mathrm{MDR}(C_{\mathrm{log}})$ be its full subcategory of motivic objects.

Weak periodic de Rham conjecture. The natural inclusion functor

$$\mathrm{MDR}(C_{\mathrm{log}})\to \mathrm{PDR}^{ss}(C_{\mathrm{log}})$$

is an equivalence. Equivalently, every irreducible periodic parabolic de Rham bundle over $C_{\mathrm{log}}$ is motivic.

This is the parabolic analogue of the periodic de Rham conjecture. The source has already established that the grading functor is fully faithful and identifies the motivic de Rham and Higgs categories; the conjecture asserts that every completely reducible periodic parabolic de Rham object is motivic.