Perfectoid $C_i$ transfer

We prove a perfectoid analogue of the Ax-Kochen theorem on zeros of $p$-adic forms: Given $d\in \mathbb{N}$, there is a finite totally ramified extension $E\mathrm{/}{\mathbb{Q}}_p$ such that every untilt of ${\mathbb{F}}_p(\text{\hspace{0.17em}⁣}(t^{1\mathrm{/}{p}^{\mathrm{\infty }}})\text{\hspace{0.17em}⁣})$ containing $E$ is $C_2(d)$. We also prove a similar result for the existence of rational points in rationally connected varieties over perfectoid field extensions of ${\mathbb{Q}}_p^{ur}$.