The conjecture on strongly topologically nilpotent crystals from rigid analytic geometry

Let $R$ be a topologically finitely generated and $p$-complete ${\mathcal{O}}_{\mathbf{C}}$-algebra, and let

$$f:R\to \mathcal{K}^+$$

be an ${\mathcal{O}}_{\mathbf C}$-homomorphism. A rational Hodge--Tate crystal on $(R/A)_{\mathbbb{\Delta}}$ is a crystal with rational Hodge--Tate structure, and its base change to $\mathcal{K}^+$ is said to be $a$-small when it belongs to the corresponding category of $a$-small crystals. The strongly topologically nilpotent crystal conjecture. For any rational Hodge--Tate crystal $\mathcal{E}$ on $(R/A)_{\mathbbb{\Delta}}$, the base change of $\mathcal{E}$ to $\mathcal{K}^+$ is $a$-small for some topologically nilpotent $a\in \mathcal{K}^\times$. This predicts that all crystals arising from rigid analytic geometry lie in the strongly topologically nilpotent class, extending the preceding classification of such crystals by generalized representations.