A Perfectoid Pro-Étale Lie Torsor Can Have a Non-Surjective Sen Map

We construct a smooth connected rigid analytic curve over ${\mathbb{C}}_p$ with an affinoid perfectoid profinite étale ${\mathbb{Z}}_p^2$-torsor whose geometric Sen morphism is not surjective, thereby disproving Rodr'ıguez Camargo’s conjectured implication from perfectoidness to surjectivity. The construction relies on a pullback theorem showing that perfectoid profinite étale towers are preserved under universally injective morphisms of rigid analytic spaces.