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

Let $C$ be an algebraically closed perfectoid field over $\Qp$ with the ring of integer $\calO_C$ and the infinitesimal thickening $\Ainf$. Let $\frakX$ be a smooth formal scheme over $\calO_C$ with a fixed smooth lifting $\wtx$ over $\Ainf$. Let $X$ be the generic fiber of $\frakX$ and $\wtX$ be its lifting over $\BdRp$ induced by $\wtx$. Let $\MIC_r(\wtX)^{{\rm H-small}}$ and $\rL\rS_r(X,\BBdRp)^{{\rm H-small}}$ be the $v$-stacks of rank-$r$ Hitchin-small integrable connections on $\wtX_{\et}$ and $\BBdRp$-local systems on $X_{v}$, respectively. In this paper, we establish an equivalence between this two stacks by introducing a new period sheaf with connection $(\calO\bB_{\dR,\pd}^+,\rd)$ on $X_{v}$.