I'm reading Laurie's note about Fargues-Fontaine Curve and I think he uses a different definition of $B_{\mathrm{cris}}$. Usually when $R$ is a perfect ring of characteristic $p$, $B^{+}_{\mathrm{cris}}(R)$ is defined as $p$-adic completion of divided power envelope of the map $W(R)\to R$ and $B_{cris}=B^{+}_{\mathrm{cris}}$.

But in these notes when $R$ is ring of valuation of an algebraically closed perfectoid field $B$ is defined as completion of $\mathrm{Frac}(W(R))$ with respect to all Gauss norms and defined Fargues-Fontaine curve by it.

I want to know the relation between $B$ and $B_{\mathrm{cris}}$ in general. Is it true that they are isomorphic if R is the valuation ring of a perfectoid field?