The Dolbeault geometric Langlands conjecture with nilpotent singular support

Let $G$ be a reductive group, let $\chi\in\pi_1(G)$ and let $w\in Z_G^{\vee}$, where $Z_G$ is the center of $G$. Let $\mathrm{Higgs}_G(\chi)$ denote the connected component of the Higgs stack corresponding to $\chi$, let $\mathcal{N}$ denote the nilpotent singular-support condition, and let $(-)_w$ denote the subcategory of central weight $w$. Write $\mathrm{B}_G$ for the relevant base of the Hitchin fibration and ${}^{L}G$ for the Langlands dual group. Dolbeault geometric Langlands conjecture. There is a $\mathrm{B}_G$-linear equivalence

$$\operatorname{IndCoh}_{\mathcal{N}}(\mathrm{Higgs}_{{}^{L}G}(w)^{\mathrm{ss}})_{-\chi} \simeq \operatorname{IndL}_{\mathcal{N}}(\mathrm{Higgs}_G(\chi))_w.$$

This is a nilpotent-singular-support version of the Dolbeault geometric Langlands correspondence, viewed as a classical limit of geometric Langlands. Its status is not resolved in the supplied text.