We revisit the model proposed earlier to account for the observed increase in the positron fraction in cosmic rays with increasing energy, in the light of new data from the Alpha Magnetic Spectrometer (AMS-02) experiment. The model accounts for the production and acceleration of secondary electrons and positrons in nearby supernova remnants which results in an additional, harder component that becomes dominant at high energies. By fitting this to AMS-02 data we can calculate the expected concomitant rise of the boron-to-carbon ratio, as well as of the fraction of antiprotons. If these predictions are confirmed by the forthcoming AMS-02 data it would conclusively rule out all other proposed explanations, in particular, dark matter annihilations or decays.