Topological $p$-adic Simpson correspondence for moduli spaces

Let $K$ be algebraically closed, let $X$ be a smooth proper rigid space over $K$, and let $n$ be a positive integer. Write $|\mathscr{B}\hspace{-0.1em}\mathrm{\textit{un}}_{\mathrm{GL}_n,v}(K)|$ for the set of isomorphism classes of rank-$n$ vector bundles on $X_v$, endowed with its natural topology, and write $|\mathscr H\hspace{-0.1em}\mathrm{\textit{iggs}}_{\mathrm{GL}_n}(K)|$ for the set of isomorphism classes of rank-$n$ Higgs bundles on $X_{{\operatorname{\acute{e}t}}}$ with its natural topology. Topological $p$-adic Simpson conjecture. The equivalence between vector bundles on $X_v$ and Higgs bundles on $X_{{\operatorname{\acute{e}t}}}$ induces a homeomorphism

$$|\mathscr{B}\hspace{-0.1em}\mathrm{\textit{un}}_{\mathrm{GL}_n,v}(K)|\xrightarrow{\sim}|\mathscr H\hspace{-0.1em}\mathrm{\textit{iggs}}_{\mathrm{GL}_n}(K)|.$$

This strengthens the $p$-adic Simpson correspondence by requiring compatibility with the natural topologies on the moduli sets, in analogy with the complex Corlette--Simpson correspondence. The supplied text does not state whether this refinement has been proved, so its status remains open here.