A $p$-adic Simpson correspondence for singular rigid-analytic varieties

Let $C$ be a complete, algebraically closed non-archimedean extension of ${\mathbb{Q}}_p$, and $X$ be a proper rigid-analytic variety over $C$. We show that the category of pro-étale vector bundles on $X$ is equivalent to the category of Higgs bundles on the $\text{eh}$-site of $X$, thereby generalizing the work of Faltings and Heuer to arbitrary proper rigid-analytic varieties.