Recent cosmic-ray (CR) antiproton studies have claimed the possibility of an excess of data over the predicted flux at around $10$ GeV, which can be the signature of dark matter annihilating into antiprotons. Nevertheless, this excess is subject to many uncertainties related to the evaluation of the antiproton spectrum produced from spallation interactions of CRs. In this work, we implement a combined Markov-chain Monte Carlo analysis of the secondary ratios of B, Be and Li and the antiproton-to-proton spectrum ($\bar{p}/p$), also including nuisance parameters to consider the uncertainties related to the spallation cross sections. This analysis allows us to constrain the Galactic halo height and the rest of propagation parameters, evaluate the impact of cross sections uncertainties in the determination of the antiproton spectrum and test the origin of the excess of antiprotons. We show that the $\bar{p}/p$ spectrum is compatible with a pure secondary origin within the uncertainties related. In particular, we find that the energy dependence of the evaluated $\bar{p}/p$ spectrum matches the AMS-02 data at energies above $3$ GeV. In particular, flat residuals at the level of $10\%$ are found with respect to the new AMS-02 $\bar{p}$dataset, which is reasonably explained by a possible underestimation on the cross sections of antiproton production, which currently constitute the main source of uncertainties in the evaluation of their flux.