Fargues–Fontaine cohomology conjecture for smooth rigid spaces

Let $C$ be an algebraically closed $p$-adic field, let $\operatorname{FF}_C$ be the Fargues–Fontaine curve with distinguished point $\infty$, and let $X$ be a quasicompact separated smooth rigid space over $C$. An overconvergent model $X^\dagger$ means a model for which the overconvergent de Rham and crystalline cohomology groups below are defined. Fargues–Fontaine cohomology conjecture. There should be a cohomology theory $H^i_{\operatorname{FF}_C}(X)$ taking values in vector bundles on $\operatorname{FF}_C$, such that, if $X$ has an overconvergent model $X^\dagger$, its fibre at $\infty$ is

$$H^i_{\mathrm{dR}}(X^\dagger/C),$$

and its completion at $\infty$ is

$$H^i_{\mathrm{crys}}(X^\dagger/B_{\mathrm{dR}}^+).$$

The conjecture extends the Fargues–Fontaine modification formalism beyond proper smooth spaces and formal models. It is related to work of Colmez–Nizioł and to the overconvergent theories of Große-Klönne; no resolution is given in the source.