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

Let $X$ be a smooth projective complex curve. The derived stack $\mathbf{Coh}_{/\mathbb A^1}^\ast(X_\mathsf{Del})$ parametrizes Deligne's $\lambda$-connections, interpolating between the Dolbeault and de Rham moduli stacks. The associated stable $\infty$-category has morphisms

$$\Phi \colon \mathsf{Coh}^{\mathsf{b}}_{\mathbb C^\ast}( \mathbf{Coh}_{/\mathbb A^1}^\ast(X_\mathsf{Del}) ) \otimes_{\mathsf{Perf}^{\mathsf{filt}}} \mathsf{Perf}_\mathbb C \longrightarrow \mathsf{Coh}^{\mathsf{b}}( \mathbf{Coh}(X_\mathsf{dR}) )$$

and

$$\Psi \colon \mathsf{Coh}^{\mathsf{b}}_{\mathbb C^\ast}( \mathbf{Coh}_{/\mathbb A^1}^\ast(X_\mathsf{Del}) ) \otimes_{\mathsf{Perf}^{\mathsf{filt}}} \mathsf{Perf}^{\mathsf{gr}} \longrightarrow \mathsf{Coh}^{\mathsf{b}}_{\mathbb C^\ast}( \mathbf{Coh}^{\mathsf{ss},0}(X_\mathsf{Dol}) ),$$

where $\mathsf{Perf}^{\mathsf{filt}}=\mathsf{Perf}([\mathbb A^1_\mathbb C/\mathbb G_m])$ and $\mathsf{Perf}^{\mathsf{gr}}=\mathsf{Perf}(\mathsf B\mathbb G_m)$. Cat-HA version of the non-abelian Hodge correspondence. The morphisms $\Phi$ and $\Psi$ are equivalences. This would identify the de Rham and Dolbeault categorified Hall algebras through the Deligne $\lambda$-connection interpolation. The supplied text does not state whether this conjecture has been proved or disproved.