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 $\calO_C$ to the case for semi-stable formal schemes. More precisely, for a liftable semi-stable formal scheme $\frakX$ over $\calO_C$ 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 $\frakX_{\et}$, 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 $\frakX_{\et}$ (equivalently, moduli stack of Higgs bundles on $X_{\et}$), generalizing the previous work of Ansch\"utz--Heuer--Le Bras.