The Lagrangian correspondence conjecture for the Dolbeault geometric Langlands correspondence

Let $\mathbb L_H(d)$ be the Lagrangian correspondence constructed between the relevant Hitchin moduli space and the Hilbert scheme of points on $T^*C$, and let the Dolbeault geometric Langlands correspondence be understood in the sense of the cited formulations. Dolbeault realization conjecture. The Lagrangian correspondence $\mathbb L_H(d)$ generically realizes the Dolbeault geometric Langlands correspondence. The claim is presented as a proposed consequence of the preceding discussion; the source mentions compatibility with known constructions and checks in special cases, but does not establish the assertion in general.