The principal 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 semisimple simply connected linear algebraic group over $F$, and let $Y$ be a principal homogeneous space under $G$ over $F$.

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

This is the second conjectured Hasse principle for semiglobal fields discussed in the paper. The abstract reports a theorem for principal homogeneous spaces under semisimple simply connected groups in the paper's stated classical-type and good-prime setting, but the full conjecture is not resolved by the supplied text.