The Fargues-Fontaine curve associated to an algebraically closed nonarchimedean field of characteristic $p$ is a fundamental geometric object in $p$-adic Hodge theory. Via the tilting equivalence it is related to the Galois theory of finite extensions of Q_p; it also occurs in Fargues's program to geometrize the local Langlands correspondence for such fields.

Recently, Peter Dillery and Alex Youcis have proposed using a related object, the "affine cone" over the aforementioned curve, to incorporate some recent insights of Kaletha into Fargues's program. I will summarize what we do and do not yet know, particularly about vector bundles on this and some related spaces (all joint work in progress with Dillery and Youcis).