$\mathbf{A}_{\text {inf}}$ has uncountable Krull dimension

Abstract: Let $\mathcal{O}E$ be a complete discrete valuation ring and $R$ be a perfect ring in characteristic $p$, we also assume $R$ is a complete valuation ring whose valuation group is of rank one and non-discrete, we prove the Krull dimension of the ring $W{\mathcal{O}_E}(R)$ of $\mathcal{O}_E$-Witt vectors over $R$ is at least the cardinality of the continuum.