The paper studies R-equivalence on smooth cubic surfaces over p-adic fields with good reduction, focusing on cases where universal (admissible) equivalence is non-trivial. The authors note that if any of these cases also had non-trivial R-equivalence, it would be strictly finer than rational (and Brauer) equivalence, contradicting a well-known conjecture by Colliot-Thélène and Sansuc.
This conjecture asserts the k-rationality of universal torsors over geometrically rational surfaces when they have a k-point. If true, it would imply the injectivity of the descent map X(k)/R → H1(k,S) for such surfaces, aligning with the triviality of Brauer equivalence in the good-reduction p-adic setting. The conjecture remains a central open problem connecting torsor rationality and arithmetic equivalence relations on rational surfaces.