Equivalence between (φ,Γ)-modules over B_{K,∞,A} and vector bundles on X_{K,A_□}^{la}/Γ_K^{la} beyond the Fredholm case

Determine whether, for a solid affinoid animated Q_{p,□}-algebra A that is not Fredholm, the natural fully faithful functor from the category of (φ,Γ_K)-modules over B_{K,∞,A} to the category of vector bundles on the quotient solid D-stack X_{K,A_□}^{la}/Γ_K^{la} is essentially surjective, hence an equivalence of categories.

Background

The paper constructs a fully faithful functor from (φ,ΓK)-modules over B{K,∞,A} to vector bundles on the locally analytic Fargues–Fontaine quotient X_{K,A_□}^{la}/Γ_K^{la}. For Banach Q_p-algebras A, this functor is proved to be an equivalence, identifying these two frameworks.

For more general coefficients A that are not Fredholm, essential surjectivity of this functor remains unsettled. Establishing equivalence in this broader context would extend the reach of the geometric interpretation of (φ,Γ)-modules and strengthen the categorical foundations of families in p-adic Hodge theory.

This question is closely tied to understanding vector bundles on solid D-stacks over less restrictive coefficient algebras and to the general behavior of the locally analytic six-functor formalism under changes of coefficients.