The AMS-02 collaboration has recently released data on the positron fraction e+/(e−+e+) up to energies of about 350 GeV. If one insists on interpreting the observed excess as a dark matter signal, then we find it is best described by a TeV-scale dark matter annihilating into τ+τ−, although this situation is already severely constrained by gamma-ray measurements.The annihilation into μ+μ− is allowed by gamma-rays more than τ+τ−, but it gives a poorer fit to AMS-02 data. Moreover, since electroweak corrections induce correlations among the fluxes of stable particles from dark matter annihilations, the recent AMS-02 data imply a well-defined prediction for the correlated flux of antiprotons. Under the assumption that their future measurements will not show any antiproton excess above the background, the dark matter interpretation of the positron rise will possibly be ruled out by only making use of data from a single experiment. This work is the first of a program where we emphasize the role of correlations among dark matter signals.