Une conjecture $C_{\rm st}$ pour la cohomologie \`a support compact

Let $\mathbf{B}$ be the ring of analytic functions on the Fargues-Fontaine curve $Y_{\rm FF}$. We show that adding $p$-adic analogs of $\log p$ and $\log 2\pi i$ kills its Galois cohomology in degrees~$\geq 1$. The analogous result for $\mathbf{B}^+_{\rm dR}$ is folklore. This makes it possible to formulate $C_{\rm dR}$ and $C_{\rm st}$-type conjectures for compact support cohomology of $p$-adic analytic varieties.