A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus

Let $C$ be an algebraically closed perfectoid field over ${\mathbb{Q}}_p$ with the ring of integer ${\mathcal{O}}_C$ and the infinitesimal thickening $\text{Ainf}$. Let $\mathfrak{X}$ be a semi-stable formal scheme over ${\mathcal{O}}_C$ with a fixed flat lifting $\tilde{\mathfrak{X}}$ over $\text{Ainf}$. Let $X$ be the generic fiber of $\mathfrak{X}$ and $\tilde{X}$ be its lifting over $\text{BdRp}$ induced by $\tilde{\mathfrak{X}}$. Let ${\text{MIC}}_r(\tilde{X}{)}^{\mathrm{H}\text{-small}}$ and $\text{rL}{\text{rS}}_r(X,\text{BBdRp}{)}^{\mathrm{H}\text{-small}}$ be the $v$-stacks of rank-$r$ Hitchin-small integrable connections on $X_{\text{et}}$ and $\text{BBdRp}$-local systems on $X_v$, respectively. In this paper, we establish an equivalence between these two stacks by introducing a new period sheaf with connection $(\text{calO}{\text{bB}}_{\text{dR},\text{pd}}^{+},\text{rd})$ on $X_v$.