Unirationality and $R$-equivalence for conic bundles over quasi-finite fields

Yanchevskiĭ had asked whether conic bundle surfaces over ${\mathbf{P}}_k^1$ are unirational when $k$ is a finite field. We give a partial answer to his question by showing that for quasi-finite fields $k$ (e.g. finite fields) a regular conic bundle $X$ over ${\mathbf{P}}_k^1$ is unirational if all non-split fibres lie over rational points. For large finite fields $k$, this beats a previous result of Mestre. Under the same assumption, we also prove that all rational points of $X$ are $R$-equivalent.