The extended Fargues--Scholze spectral action

Let $G$ be a connected reductive group over a non-archimedean local field. The main result of Fargues and Scholze [FS21] for the geometrization of the local Langlands correspondence is the construction of a ``spectral action'' on the category of $\ell$-adic sheaves on $\text{Bun}_{G}$, the stack of $G$-torsors on the Fargues-Fontaine curve. The goal of this paper is to prove a conjecture of Fargues which says that one can extend this construction to the larger stack $\text{Bun}_G^e$ of $G$-torsors on the Kaletha gerbe over the curve, as introduced by Fargues [Far22]. This ``extended spectral action'' allows for a version of the categorical local Langlands conjecture for an arbitrary connected reductive group $G$, and is the first such statement for those $G$ which are not extended pure inner forms of a quasi-split group, such as non-trivial inner forms of $\mathrm{SL}_{n}$. Finally, we prove this conjecture for tori, following the original argument of Zou [Zou24] and, when the center $Z_{G}$ of $G$ is connected and $H^{1}(F,Z_{G})=0$, we reduce the ``extended'' version of the categorical conjecture to the one in Fargues--Scholze.