The categorical arithmetic local Langlands conjecture

Assume that $G$ is quasi-split over $F$, equipped with a pinning $(B,T,e)$, and fix a non-trivial additive character $\psi:F\to {\mathbb Z}_\ell[\mu_{p^\infty}]^\times$. Let $\Loc_{\widehat{G}}$ be the stack of local Langlands parameters and let $\Shv_c({\mathfrak B}(G),\Lambda)$ be the category of constructible sheaves on the stack of $G$-bundles on the Fargues-Fontaine curve, with coefficients in $\Lambda$. The categorical arithmetic local Langlands conjecture. There is a canonical equivalence of categories

$${\mathbb L}_G:\Coh(\Loc_{\widehat{G}}\otimes\Lambda)\cong\Shv_c({\mathfrak B}(G),\Lambda).$$

This is presented as a categorical form of arithmetic local Langlands, inspired by the global geometric Langlands conjecture; related formulations replace the constructible category by a larger sheaf category and impose support conditions.