Emerton–Gee–Hellmann's categorical p-adic local Langlands conjecture

Let $K$ be a finite extension of $\mathbb{Q}_p$. Write $\mathcal{D}(\ast/GL_n(K)^{\la})$ for the stable $\infty$-category of locally analytic representations of $GL_n(K)$, let $\mathfrak{X}_{n,K}$ be the rigid analytic moduli stack of $G_K$-equivariant vector bundles over the Fargues–Fontaine curve $X_{\mathbb{C}_p}$, and let $\mathcal{D}(\mathfrak{X}_{n,K})$ be the stable $\infty$-category of solid quasi-coherent sheaves on $\mathfrak{X}_{n,K}$. Emerton–Gee–Hellmann's categorical p-adic local Langlands conjecture. There exists an exact functor

$$\mathfrak{A}_{GL_n(K)}^{\rig} \colon \mathcal{D}(\ast/GL_n(K)^{\la}) \to \mathcal{D}(\mathfrak{X}_{n,K})$$

satisfying many good properties, including compatibilities with other categorical local Langlands correspondences. This is a $p$-adic analogue of the categorical local Langlands correspondence and is intended to geometrize the relation between locally analytic representations and $G_K$-equivariant vector bundles; the conjecture remains open in the stated generality.