Overconvergence of étale $(\phi ,\mathrm{\Gamma })$-modules in families

We prove a conjecture of Emerton, Gee and Hellmann concerning the overconvergence of étale $(\phi ,\mathrm{\Gamma })$-modules in families parametrized by topologically finite type ${\mathbb{Z}}_p$-algebras. As a consequence, we deduce the existence of a natural map from the rigid fiber of the Emerton-Gee stack to the rigid analytic stack of $(\phi ,\mathrm{\Gamma })$-modules.