Torsion finiteness conjecture for smooth affinoid rigid spaces

Let $C$) be an algebraically closed complete non-archimedean extension of $\bQ_p$, with ring of integers $\bO_C$. Let $X$ be a smooth affinoid rigid space over $C$.

Torsion finiteness conjecture.

1. For all $i\in \mathbb{Z}$, the torsion subgroup of

$$H^i_{\operatorname{pro\acute{e}t}}(X, \mathbb{Z}_p)$$

is finite. 2. For all $n\in \mathbb{Z}_{\ge 0}$, the $p^n$-torsion subgroup of $\operatorname{Pic}(X)$ is finite.

The conjecture concerns integral $p$-adic pro-étale cohomology and Picard groups of non-proper rigid-analytic spaces, whose rational cohomology need not be finite-dimensional. The source provides no resolution status.