Emerton–Gee–Hellmann overconvergence conjecture for étale $(\varphi,\Gamma)$-modules

Let $K$ be a finite extension of $\mathbf{Q}_p$, let $A$ be a $p$-adically complete, topologically of finite type $\mathbf{Z}_p$-algebra, and let $\mathrm{{\bf A}}_{K,A}^{\dagger}$ be the ring of overconvergent periods naturally contained in both $\mathrm{{\bf A}}_{K,A}$ and the Robba ring $\mathcal{R}_{K,A}$. An étale $(\varphi,\Gamma)$-module over $\mathrm{{\bf A}}_{K,A}$ is the object considered in the moduli stack $\mathcal{X}_d$ of projective étale $(\varphi,\Gamma)$-modules.

Emerton–Gee–Hellmann overconvergence conjecture. Every étale $(\varphi,\Gamma)$-module over $\mathrm{{\bf A}}_{K,A}$ canonically descends to an étale $(\varphi,\Gamma)$-module over $\mathrm{{\bf A}}_{K,A}^{\dagger}$. Consequently, the map

$$\pi_d:\mathcal{X}_d^{\mathrm{rig}}\longrightarrow\mathfrak{X}_d$$

exists.

This conjecture supplies the missing link between the moduli stack of étale $(\varphi,\Gamma)$-modules and the rigid analytic stack of $(\varphi,\Gamma)$-modules, by allowing one to forget the étale lattice through the overconvergent period ring. The paper states that it proves this conjecture and deduces the existence of the map $\pi_d$.