The quantized Lagrangian correspondence conjecture for the de Rham geometric Langlands correspondence

Let $\mathbf{L_H(d)}$ be the Fourier transform associated with the Lagrangian correspondence $\mathbb L_H(d)$, and let $\mathbf{q\text{-}L_H(d)}$ be its quantization to a functor

$$\mathbf{q\text{-}L_H(d)}:\operatorname{\mathcal D\text{-}mod}(\operatorname{Hilb}^d(C))\longrightarrow\operatorname{\mathcal D\text{-}mod}(\operatorname{Bun}_{n,\Lambda}).$$

Here $\operatorname{Bun}_{n,\Lambda}$ is the moduli space of rank-$n$ semistable bundles with fixed determinant $\Lambda$. De Rham realization conjecture. The Fourier transform $\mathbf{L_H(d)}$ can be quantized to the functor $\mathbf{q\text{-}L_H(d)}$ above, and, for sufficiently large $d$, $\mathbf{q\text{-}L_H(d)}$ realizes the de Rham geometric Langlands correspondence in the sense of Drinfeld. The source motivates this by the expected image of a flat-bundle input and gives no general proof of the realization.