The Brauer–Manin sufficiency conjecture for rationally connected surfaces over finite-field function fields

Let $C$ be a nice curve over a finite field~$\mathbf{F}$, and let $K$ be its function field. Let $X$ be a proper, smooth, geometrically integral and separably rationally connected surface over~$K$. Write (BM) for the Brauer–Manin obstruction condition, namely that the Brauer–Manin obstruction is the only obstruction to the existence of a $K$-rational point on $X$.

Colliot-Thélène–Sansuc analogue. The surface $X$ verifies (BM).

This is an analogue, in positive characteristic, of a conjecture of Colliot-Thélène and Sansuc concerning the sufficiency of the Brauer–Manin obstruction over number fields. The source presents it as an open question and uses it to study unirationality of conic bundles over the projective line over finite fields.