Current measurements of cosmic-ray fluxes have reached unprecedented accuracy thanks to the new generation of experiments, and in particular the AMS-02 mission. At the same time, significant progress has been made in the propagation models of galactic cosmic rays. These models include several propagation parameters, which are usually inferred from the ratios of secondary to primary cosmic rays, and which depend on the cross sections describing the collisions among the various species of cosmic-ray nuclei. At present, our knowledge of these cross sections in the energy range where cosmic-ray interactions occur is limited, and this is a source of uncertainties in the predicted fluxes of secondary cosmic-ray nuclei. In this work we study the impact of the cross section uncertainties on the fluxes of light secondary nuclei (Li, Be, B) using a preliminary version of the upcoming {tt DRAGON2} code. We first present a detailed comparison of the secondary fluxes computed by implementing different parametrizations for the network of spallation cross sections. Then, we propose for the first time the use of secondary-over-secondary cosmic-ray flux ratios as a tool to investigate the consistency of cross sections models and give insight of the overall uncertainties coming from the cross sections parametrizations. We show that the uncertainties inferred from the cross section data are enough to explain the discrepancies in the Be and Li fluxes with respect to the AMS-02 data, with no need of a primary component in their spectra. In addition, we show that the fluxes of B, Be and Li can be simultaneously reproduced by rescaling their cross sections within the experimental uncertainty. Finally, we also revisit the diffusive estimation of the halo size, obtaining good agreement with previous works and a best fit value of $6.8 pm 1$ kpc from the most updated cross sections parametrizations.
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