Colliot-Thélène–Parimala–Suresh Hasse principle conjecture for projective homogeneous spaces

Let $k$ be a $p$-adic field, and let $F$ be the function field of a smooth projective geometrically integral curve defined over $k$. Let $\Omega$ denote the set of discrete (rank 1) valuations on $F$ which either extend the norm of $k$ or are trivial on $k$. Let $G/F$ be a connected linear algebraic group, and let $X/F$ be a projective homogeneous space over $G$. For each $v\in\Omega$, write $F_v$ for the completion of $F$ with respect to $v$. Colliot-Thélène–Parimala–Suresh's conjecture. The Hasse principle should hold for $X$, namely

$$X(F)\neq\emptyset \iff X(F_v)\neq\emptyset\quad\forall v\in\Omega.$$

This conjecture proposes a Hasse principle for projective homogeneous spaces over function fields of curves over $p$-adic fields. The paper develops an analytic approach to such local-global questions and proves the principle for certain families, while the general assertion remains open in the supplied context.