On smooth adic spaces over BdR⁺ and sheafified p-adic Riemann--Hilbert correspondence

Inspired by Heuer's paper, we consider adic spaces over a de Rham period ring of a perfectoid Tate--Huber ring and their sheafified Riemann--Hilbert correspondence. We will prove that for any smooth adic space X over BdR,α⁺(K,K⁺), there is a canonical sheaf isomorphism R¹ν_*(GLᵣ(BdR⁺,X̄/t^α)) t-MICᵣ(X). Moreover, we will define v-prestacks of BdR⁺-local systems and t-connections, and prove that they are small v-stacks.