A stacky nilpotent $p$-adic Riemann-Hilbert correspondence

Let $\bar{X}$ be a smooth rigid variety over $C={\mathbb{C}}_p$ admitting a lift $X$ over $B_{dR}^{+}$. In this paper, we use the stacky language to prove a nilpotent $p$-adic Riemann-Hilbert correspondence. After introducing the moduli stack of ${\mathbb{B}}_{dR}^{+}$-local systems and $t$-connections, we prove that there is an equivalence of the nilpotent locus of the two stacks: $R{H}^0:L{S}_X^0\to tMI{C}_X^0$, where $L{S}_X^0$ is the stack of nilpotent ${\mathbb{B}}_{dR}^{+}$-local systems on ${\bar{X}}_{1,v}$ and $tMI{C}_X^0$ is the stack of ${\mathcal{O}}_X$-bundles with integrable $t$-connection on $X_{et}$.