The projective homogeneous space Hasse principle over p-adic curve function fields

Let $F$ be the function field of a curve over a $p$-adic field, and let $\Omega_F$ be the set of all discrete valuations of $F$. For each $\nu\in\Omega_F$, let $F_\nu$ be the completion of $F$ at $\nu$. Let $G$ be a connected linear algebraic group over $F$, and let $Y$ be a projective homogeneous space under $G$.

Projective homogeneous space Hasse principle. If $Y(F_\nu)\neq\varnothing$ for every $\nu\in\Omega_F$, then $Y(F)\neq\varnothing$.

This is one of two proposed analogues of the Hasse principle for semiglobal fields. The paper's abstract proves the corresponding local-global principle for groups of classical type when the residue characteristic is good, but the conjecture is stated in greater generality and its resolution is not established by the supplied text.