The affinoid–analytic equality for multivariate Robba rings

Let $G$ act continuously on $\tilde{\mathbf{A}}^{\dagger,r}_{L_\infty}$, and let $\tilde{\mathbf{A}}^{\dagger,r,\operatorname{aff}}_{K_\infty}$ be the subring of elements admitting an affinoid model, while $\tilde{\mathbf{A}}^{\dagger,r,\operatorname{an}}_{K_\infty}$ is the subset of uniformly $r$-analytic elements for the action of $G$. Affinoid–analytic equality. One has

$$\tilde{\mathbf{A}}^{\dagger,r,\operatorname{aff}}_{K_\infty}=\tilde{\mathbf{A}}^{\dagger,r,\operatorname{an}}_{K_\infty}.$$

The inclusion from left to right is known; the conjecture asserts the converse. Its difficulty is the lack of an obvious construction of $G$-stable affinoid models, although such models can be constructed when $K_\infty$ is the division field of a formal group, with possible extensions to certain arithmetic dynamical systems.