$p$-adic non-abelian Hodge theory for curves via moduli stacks

For a smooth projective curve $X$ over $\mathbb C_p$ and any reductive group $G$, we show that the moduli stack of $G$-Higgs bundles on $X$ is a twist of the moduli stack of v-topological $G$-bundles on $X_v$ in a canonical way. We explain how a choice of an exponential trivialises this twist on points. This yields a geometrisation of Faltings' $p$-adic Simpson correspondence for $X$, which we recover as a homeomorphism between the points of moduli spaces. We also show that our twisted isomorphism sends the stack of $p$-adic representations of $\pi_1(X)$ to an open substack of the stack of semi-stable Higgs bundles of degree $0$.