Derived Langlands equivalence for parabolic connections via the partially compactified Radon transform

Let $\mathcal{M}$ be the moduli stack of parabolic connections, let $P^{\vee}$ be the dual projective space, and let $D_{\boldsymbol\nu}$ be the corresponding twisted differential-operator ring. Write $\mathcal{D}_{qc}(\mathcal{M})^-$ for the component on which $-1\in\mu_2$ acts as $-1$ on every cohomology sheaf. Derived Langlands equivalence via the partially compactified Radon transform. The connected component of the derived category of quasi-coherent sheaves on $\mathcal{M}$ is equivalent to the derived category of $D_{\boldsymbol\nu}$-modules on $P^{\vee}$:

$$L'\colon\mathcal{D}_{qc}(\mathcal{M})^-\xrightarrow{\sim}\mathcal{D}(P^{\vee},D_{\boldsymbol\nu}).$$

The source presents this as an equivalent formulation conditional on the partially compactified Radon transform conjecture, so it remains open with that conjecture in the supplied text.