Moduli stacks of generalized phi-modules

Let $F$ be an arbitrary $p$-adic field and let $G$ be an arbitrary reductive group over $F$ with Langlands dual group $^LG$. We show that the change-of-group morphism of Emerton-Gee stacks $\mathcal{X}_{^LG}\to\mathcal{X}_{GL_d}$ is relatively representable by algebraic stacks of finite presentation over $\operatorname{Spf}\mathbf{Z}_p$ for any embedding $^LG\to GL_d$, which improves the result of \cite{Min25} which says the morphism is representable by locally Noetherian formal algebraic stacks.