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In his 2012 CDM proceedings, Peter Scholze mentions the following open question:

Let $K$ be a perfectoid field and $(A,A^+)$ a complete affinoid $K$-algebra. Suppose there exists a cover of $X = \text{Spa}(A,A^+)$ by rational subsets $U_i \subset X$ such that $\mathcal{O}_X(U_i)$ is a perfectoid $K$-algebra. Does it follow that $A$ is a perfectoid $K$-algebra?

Has there been any progress on this question since then?

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    $\begingroup$ I'm not sure about the most current status of this, but for a start you can look at Buzzard & Verberkmoes "Stably Uniform Affinoids are Sheafy", where they prove the result if $A$ is stably uniform and of characteristic $p$. $\endgroup$
    – Alex Mathers
    Commented Nov 19, 2019 at 8:40
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    $\begingroup$ This is still an open problem. $\endgroup$
    – David Hansen
    Commented Nov 19, 2019 at 11:12

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