Explicit local Langlands conjecture for toral supercuspidal representations

Let $G$ be a connected simple algebraic group, let $\hg$ be the associated affine Kac–Moody algebra at critical level, and let $(J,\tphi)$ be a $K$-type for the loop group obtained from a Yu datum, with Lie algebra $\fj$. Define the central support $\Op_{\Lg}(\fj,\tphi)$ from the induced representation $\Vac_{\fj,\tphi}$. Explicit local geometric Langlands conjecture. (i) There exists a unique irreducible formal $\LG$-connection $\nabla_{\tphi}$ such that

$$\Op_{\Lg}(\fj,\tphi)=p^{-1}(\nabla_{\tphi}).$$

(ii) Yu data inducing isomorphic supercuspidal representations give $K$-types with the same central support and isomorphic corresponding connections. (iii) Every irreducible formal $\LG$-connection arises as some $\nabla_{\tphi}$. (iv) This correspondence coincides with the supercuspidal $L$-packets constructed by Kaletha. The formulation is intended to make the local geometric Langlands correspondence explicit; part (i) has been essentially proved for epipelagic representations, while the full correspondence remains open.