Categorical local Langlands conjecture for elliptic parameters

Let $G$ be quasisplit over $E$, fix a Whittaker datum, and assume that the connected split center of $G$ is trivial. Let $\varphi$ be an elliptic $L$-parameter, let $S_\varphi$ be its component group, and let $C_\varphi$ denote the corresponding spectral component. Elliptic categorical local Langlands conjecture. There is a unique generic supercuspidal representation $\pi$ of $G(E)$ with $L$-parameter $\varphi_\pi=\varphi$, and the functor

$$\operatorname{Perf}([\operatorname{Spec}\overline{\mathbb Q}_\ell/S_\varphi])\to\mathcal D_{\mathrm{lis}}^{C_\varphi}(\operatorname{Bun}_G,\overline{\mathbb Q}_\ell)^\omega, \qquad W\longmapsto\operatorname{Act}_W(\pi)$$

is an equivalence. Consequently, irreducible supercuspidal representations of some $G_b(E)$ with parameter $\varphi$ are in bijection with irreducible representations of $S_\varphi$, and the equivalence is $t$-exact for the standard $t$-structures. This is the elliptic specialization of the categorical local Langlands conjecture, with the additional $t$-exactness assertion.