The overconvergence of multivariable $({\phi }_q,{\mathcal{O}}_K^{\times })$-modules at the perfectoid level

Let $K$ be a finite unramified extension of ${\mathbb{Q}}_p$, and $E$ a finite extension of $K$ with ring of integers ${\mathcal{O}}_E$. We define the overconvergence of multivariable $({\phi }_q,{\mathcal{O}}_K^{\times })$-modules over $A_{\mathrm{m}\mathrm{v},E}$ and explore some basic properties. We prove the overconvergence at the perfectoid level using the geometry of relative Fargues-Fontaine curve.