Rodríguez Camargo’s conjectured implication from perfectoidness to Sen-map surjectivity

Rodríguez Camargo’s conjecture asserts that perfectoidness of the relevant torsor forces surjectivity of its Sen map. The latest claim gives a smooth connected rigid analytic curve over $\mathbb{C}_p$ with a perfectoid torsor whose Sen morphism is non-surjective.

Known results

• Abelian varieties: a January 2025 preprint claims the conjecture is completely resolved, identifying perfectoidness with injectivity of the geometric Sen morphism, equivalently surjectivity of the dual map.
• The same criterion is claimed for semi-abelian varieties and $p$-divisible rigid analytic groups under stated hypotheses.
• Sean Howe’s December 2025 preprint reports the conjecture as known for semi-abelian bases with compact $K$, while broader global perfectoidness remains expected.
• July 2024 work proves related stalkwise perfectoidness results, not the general curve case.

September 2026 claimed counterexample

An arXiv preprint reported on September 8, 2026 claims the explicit curve example and therefore would disprove the general implication. The claim is unrefereed and has no independent confirmation in the retrieved sources.

Current status (as of September 2026): Special abelian and semi-abelian cases are claimed resolved, while the general conjecture and the proposed curve counterexample remain unverified.