If the weight, but not the inertial mass, of an antiproton should differ from that of a proton, there would be important consequences for the theories of relativity and of particles. But informed opinion is sceptical of the possibility.
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Maddox, J. Weak equivalence in the balance. Nature 350, 187 (1991). https://doi.org/10.1038/350187a0
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DOI: https://doi.org/10.1038/350187a0