Perfectoidness criterion via tangent spaces

Determine whether the following conjectural perfectoidness criterion holds: for a K-torsor \tilde{Z} over a smooth rigid analytic variety Z over a p-adic field L, with base change to C = \overline{L}^\wedge, prove that \tilde{Z}_C is perfectoid whenever, for every geometric point \tilde{z}, the Banach–Colmez tangent space T_{\tilde{z}}\tilde{Z} = \overline{T_{\tilde{z}}\tilde{Z}^} is perfectoid. Equivalently, establish the conjecture formulated in terms of the geometric Sen morphism that characterizes perfectoidness of \tilde{Z}_C via perfectoidness of all tangent spaces.

Background

The paper introduces inscribed v-sheaves and computes tangent bundles and derivatives of period maps for torsors \tilde{Z}/Z, providing a differential framework in which Banach–Colmez tangent spaces can be analyzed. In this context, the author proposes a criterion relating perfectoidness of the entire torsor to perfectoidness of the tangent spaces at all geometric points.

The conjecture aligns with and is stated to be equivalent to a conjecture of Rodríguez Camargo formulated via the geometric Sen morphism, for which there has been recent progress. Establishing this criterion would connect local differential properties (tangent spaces) to global geometric properties (perfectoidness), offering a practical test for perfectoidness using the classification of perfectoid Banach–Colmez spaces.