Integral Brauer–Manin conjecture for log rationally connected varieties

Let $k$ be a number field, let $S$ be a finite set of places of $k$, and let $\mathcal{O}_S$ be the ring of $S$-integers. An $\mathcal{O}_S$-scheme is $S$-split when its base change to $k_v$ is split for at least one $v \in S$. Let $\mathcal{X}$ be an $S$-split smooth $\mathcal{O}_S$-scheme, and set

$$X = \mathcal{X} \otimes_{\mathcal{O}_S} k.$$

Assume that $X$ is a simply connected, log rationally connected variety. Let $\mathcal{X}(\mathbb{A}_{k,S})^{\operatorname{Br}(X)}$ denote the $S$-integral adelic points surviving the Brauer–Manin pairing.

Integral Brauer–Manin conjecture. If

$$\mathcal{X}(\mathbb{A}_{k,S})^{\operatorname{Br}(X)} \neq \emptyset,$$

then

$$X(\mathcal{O}_S) \neq \emptyset.$$

This is proposed as an integral analogue of the Colliot-Thélène–Sansuc conjecture. The paper proves that the integral Brauer–Manin obstruction is not always the only obstruction for $S$-split log K3 surfaces, but expects the displayed implication for the narrower class of simply connected log rationally connected schemes.