The Alpha Magnetic Spectrometer (AMS-02) experiment has ushered cosmic-ray physics into precision era. In an earlier paper [Génolini et al., Phys. Rev. D 99, 123028 (2019), 10.1103/PhysRevD.99.123028], we designed an improved method to calibrate propagation models on the boron-to-carbon ratio data. Here we provide a robust prediction of the p ¯ flux, accounting for several sources of uncertainties and their correlations. Combined with a correlation matrix for the p ¯ data, we show that the latter are consistent with a secondary origin. This paper presents key elements relevant to the dark matter search in this channel, notably by pointing out the inherent difficulties in achieving predictions at the percent-level precision.