The J/ψ meson has negative G parity so that, in the limit of isospin conservation, its decay into π+π− should be purely electromagnetic. However, the measured branching fraction B(J/ψ→π+π−) exceeds by more than 4.5 standard deviations the expectation computed according to BABAR data on the e+e−→π+π− cross section. The possibility that the two-gluon plus one-photon decay mechanism is not suppressed by G-parity conservation is discussed, even by considering other multipion decay channels. As also obtained by phenomenological computation, such a decay mechanism could be responsible for the observed discrepancy. Finally, we notice that the BESIII experiment, having the potential to perform an accurate measurement of the e+e−→π+π− cross section in the J/ψ mass energy region, can definitely prove or disprove this strong G-parity-violating mechanism by confirming or confuting the BABAR data.