A $p$-adic Simpson correspondence for smooth proper rigid varieties

For any smooth proper rigid analytic space $X$ over a complete algebraically closed extension of ${\mathbb{Q}}_p$, we construct a $p$-adic Simpson correspondence: an equivalence of categories between vector bundles on Scholze’s pro-étale site of $X$ and Higgs bundles on $X$. This generalises a result of Faltings from smooth projective curves to any higher dimension, and further to the rigid analytic setup. The strategy is new, and is based on the study of rigid analytic moduli spaces of pro-étale invertible sheaves on spectral varieties.