On the cohomology of local Shimura varieties

Let $G$ be a reductive group over $Q_p$. Fargues conjectured a geometric Langlands type statement
for the stack Bun_G of $G$-bundles on the Fargues-Fontaine curve, refining the conjectural local
Langlands correspondence. The conjecture requires a formalism of "constructible" $l$-adic sheaves
on Bun_G, and a geometric Satake equivalence for the B_dR-Grassmannian. We will explain some
progress towards establishing such a formalism. If time permits, we will indicate some expected
applications of the formalism to open problems in the area.