Pro-étale uniformisation conjecture for curves

Let $K$ be the base field, and let $C$ be a connected smooth projective curve over $K$ of genus $g\geq 1$. In the diamantine setting, one expects a universal pro-étale perfectoid cover $\widetilde{I}_C\to C$ and, for a choice of base point $x\in C(K)$, a uniformisation

$$C^\diamondsuit=\widetilde{I}_C/\pi_1(C,x),$$

where $\pi_1(C,x)$ is the étale fundamental group. Pro-étale uniformisation conjecture. The isomorphism class of $\widetilde{I}_C$ is locally constant in the moduli space $\mathcal M_g(K)$ of connected smooth projective curves of genus $g$ over $K$ with its non-archimedean topology. This extends the known abeloid and elliptic-curve uniformisations toward a non-archimedean analogue of complex uniformisation; the claim is presented as conjectural in the source, and its status for general curves is unresolved.