Geometric Sen perfectoidness conjecture

Let $X$ be an fs log smooth adic space over $(C,C^+)$ with normal crossing divisors. Let $G$ be a compact $p$-adic Lie group, and let $\widetilde{X}\to X$ be a pro-Kummer-étale $G$-torsor. The geometric Sen operator is the morphism

$$\theta_{\widetilde{X}}:(\operatorname{Lie} G)^{\vee}_{\mathrm{k\acute{e}t}}\otimes_{\widehat{\mathbb{Q}}_p}\widehat{\mathscr{O}}_X\to \Omega_X^1(\log)\otimes_{\mathscr{O}_X}\widehat{\mathscr{O}}(-1).$$

Geometric Sen perfectoidness conjecture. The morphism $\theta_{\widetilde{X}}$ is surjective if and only if $\widetilde{X}$ is a perfectoid space.

This conjecture proposes that the geometric Sen operator detects precisely the directions in which $X$ becomes perfectoid. It is presented as a geometric generalization of Sen's theorem that a $p$-adic Galois representation of a $p$-adic field has vanishing Sen operator if and only if it is potentially unramified; the conjecture's resolution is not specified here.