Let $p: \mathcal{X} \rightarrow \text{Spec } \mathcal{O}_K$ be a normal proper Artin stack with finite diagonal. A $K$-rational point is by definition a section $x: \text{Spec}(K) \rightarrow \mathcal{X}$ of $p$ over the generic point of $\text{Spec } \mathcal{O}_K$, and an integral point is a section $\bar{x}: \text{Spec } \mathcal{O}_K \rightarrow \mathcal{X}$ of $p$.
Now if $\mathcal{X}$ were to be a proper scheme, then every rational point extends uniquely to an integral point. Indeed, the valuative criterion gives morphisms from every division ring that we can patch up to a morphism from $\text{Spec } \mathcal{O}_K$.
I am currently reading the work "Heights on stacks and a generalized Batyrev–Manin–Malle conjecture" and there it is claimed that the above statement is no longer true for stacks, but it is still true after a (possibly) ramified extension of $\text{Spec } \mathcal{O}_K$. This leads to my two questions:
- Why is this not true for stacks? Although a specific counterexample would be great, I am also happy with some intuition why this ought to fail for stacks.
- Why is it still true after a (possibly) ramified extension of $\text{Spec } \mathcal{O}_K$? In this case I am mostly interested in a precise argument.