Wittenberg's descent conjecture for torsors under linear algebraic groups

Let $k$ be a number field, let $\Omega$ be the set of places of $k$, and write $k_{\Omega}=\prod_{v\in\Omega}k_v$. For a smooth $k$-variety $X$, let $X(k_{\Omega})^{\operatorname{Br}_{\mathrm{nr}}(X)}$ denote the left kernel of the Brauer–Manin pairing. Let $G$ be a linear algebraic $k$-group, let $f:Y\to X$ be a $G$-torsor, and for each $[\sigma]\in H^1(k,G)$ let ${_{\sigma}}Y$ be the corresponding twist of $Y$.

Wittenberg's descent conjecture. Let $X$ be a smooth $k$-variety and let $G$ be a linear algebraic $k$-group. Let $f:Y\to X$ be a $G$-torsor with $Y$ rationally connected. Assume that

$${_{\sigma}}Y(k)\text{ is dense in }{_{\sigma}}Y(k_{\Omega})^{\operatorname{Br}_{\mathrm{nr}}({_{\sigma}}Y)}$$

for any $[\sigma]\in H^1(k,G)$. Then

$$X(k)\text{ is dense in }X(k_{\Omega})^{\operatorname{Br}_{\mathrm{nr}}(X)}.$$

This is Wittenberg's descent principle: weak approximation with Brauer–Manin obstruction for every twist of a rationally connected torsor should descend to the base. The supplied source presents it as a consequence of the broader study of rational points and as the main conjecture addressed by the paper; no resolution status is supplied here.