The small $p$-adic Simpson correspondence in the semi-stable reduction case

We generalize several known results on small Simpson correspondence for smooth formal schemes over ${\text{calO}}_C$ to the case for semi-stable formal schemes. More precisely, for a liftable semi-stable formal scheme $\text{frakX}$ over ${\text{calO}}_C$ with generic fiber $X$, we establish (1) an equivalence between the category of Hitchin-small integral $v$-bundles on $X_v$ and the category of Hitchin-small Higgs bundles on ${\text{frakX}}_{\text{et}}$, generalizing the previous work of Min–Wang, and (2) an equivalence between the moduli stack of $v$-bundles on $X_v$ and the moduli stack of rational Higgs bundles on ${\text{frakX}}_{\text{et}}$ (equivalently, moduli stack of Higgs bundles on $X_{\text{et}}$), generalizing the previous work of Anschütz–Heuer–Le Bras.