Antieau–Williams conjecture on failure of purity for non-special semisimple groups

Let $k$ be a field, let $G$ be a connected reductive group over $k$, and let $X$ be an irreducible smooth $k$-scheme with function field $K$. Purity for $G$-torsors over $X$ means that the map

$$\operatorname{im}\left({\mathrm H}^1_{\mathrm{\acute{e}t}}(X,G)\to {\mathrm H}^1_{\mathrm{\acute{e}t}}(\operatorname{Spec} K,G)\right)\to \bigcap_{x\in X^{(1)}}\operatorname{im}\left({\mathrm H}^1_{\mathrm{\acute{e}t}}(\operatorname{Spec} \mathcal{O}_{X,x},G)\to {\mathrm H}^1_{\mathrm{\acute{e}t}}(\operatorname{Spec} K,G)\right)$$

is surjective. Antieau–Williams conjecture. If $G$ is a non-special, semisimple $k$-group scheme, then there exists a smooth, affine $k$-scheme for which purity fails. The conjecture predicts that purity for torsors is not a general phenomenon beyond the special case; the supplied text gives no resolution status, so it remains open here.