The filtered Cat-HA version of the non-abelian Hodge correspondence

Let $X$ be a smooth projective complex curve. The pro-completed category $\mathsf{Coh}^{\mathsf b}_{\mathsf{pro},\mathbb C^\ast}(\mathbf{Coh}^{(\ast)}_{/\mathbb A^1}(X_\mathsf{Del}))$ is identified with the corresponding category over $[\mathbb A^1/\mathbb G_m]$ and is a module over $\mathsf{Perf}^{\mathsf{filt}}$. Filtered Cat-HA version of the non-abelian Hodge correspondence. The morphisms $\Phi^\ast$ and $\Psi^\ast$ are equivalences; equivalently, $\mathsf{Coh}^{\mathsf b}_{\mathsf{pro}}(\mathbf{Coh}(X_\mathsf{dR}))$ is filtered by $\mathsf{Coh}^{\mathsf b}_{\mathsf{pro},\mathbb C^\ast}(\mathbf{Coh}^\ast_{/\mathbb A^1}(X_\mathsf{Del}))$ with associated graded $\mathsf{Coh}^{\mathsf b}_{\mathsf{pro},\mathbb C^\ast}(\mathbf{Coh}^{\mathsf{ss},0}(X_\mathsf{Dol}))$. Following Simpson, this is expected to express the filtered relationship between de Rham and Dolbeault categorified Hall algebras, but the supplied text gives no resolution result.