The p-adic non-abelian Hodge correspondence for smooth projective rigid analytic spaces

Let $X$ be a smooth projective rigid analytic space over $\mathbb{C}_p$, and let $G$ be a reductive group defined over $\mathbb{C}_p$. A $G(\mathbb{C}_p)$-representation of $\pi_1^{\mathrm{et}}(X)$ is a representation of the étale fundamental group of $X$ in $G(\mathbb{C}_p)$, while a semistable $G$-Higgs bundle with vanishing Chern classes is the corresponding Higgs-theoretic object. The p-adic non-abelian Hodge conjecture. There is an equivalence of categories

$$\{G(\mathbb{C}_p)\text{-representations of }\pi_1^{\mathrm{et}}(X)\}\longleftrightarrow\{\text{semistable $G$-Higgs bundles with vanishing Chern classes}\}.$$

The conjecture is proposed as an ingredient for understanding the p-adic anabelian problem in the source; the supplied text gives no resolution status.