Wilson–Hecke compatibility in Dolbeault geometric Langlands

Let $C$ be a smooth projective curve, let $G$ be a reductive group, and let $^{L}G$ be its Langlands dual. For a minuscule coweight $\mu$ and a point $x\in C$, let $W_x^\mu$ be the Wilson operator on the coherent category of semistable $^{L}G$-Higgs bundles and let $H_x^\mu$ be the corresponding Hecke operator on the limit category of $G$-Higgs bundles. Wilson–Hecke compatibility conjecture. The Dolbeault geometric Langlands equivalence

$$\operatorname{Coh}(\mathrm{Higgs}_{^{L}G}(w)^{\mathrm{ss}})_{-\chi}\simeq\operatorname{L}(\mathrm{Higgs}_{G}(\chi))_w$$

intertwines $W_x^\mu$ and $H_x^\mu$ for every minuscule coweight $\mu$. This is the expected operator-level refinement of the main equivalence; no resolution is given.