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In 1954, Ashkin, Page and Woodward (hereafter APW) reported on the first counter experiments to measure the electron-electron and electron-positron scattering cross-sections at low relativistic energies (0.6-1.7 MeV). Their aim was to look for the spin and exchange or virtual annihilation effects predicted by the Moller and Bhabha formulae. Their experiments confirmed these effects, but the measured cross-sections at 0.61 MeV were significantly smaller than their predicted values. The authors remarked that these deviations were `presumably due to multiple scattering'. However, careful reading of the unpublished theses of Page (1950) and Ashkin (1952), Page's letter (1951) and the APW paper reveals no credible evidence of multiple scattering at 0.61 MeV; if anything, the evidence rules against multiple scattering. If multiple scattering is ruled out, the observations may indicate a departure from quantum electrodynamics. This departure may be due to a non-Coulomb central force, a weakening of the spin-statistics connection, or both. Only experiment can tell which of these possibilities holds true, and therefore we suggest the new electron-electron scattering experiments be carried out at different energies (at 0.4-1.0 MeV) and different scattering angles, as well as specific tests for multiple scattering. We consider a non-Coulomb central force to be very unlikely, and advance the hypothesis that a fraction of the electron pairs scatter as spin-zero fermions (which would lower the observed cross-section). Numerical calculations show that the hypothesis may be tested quantitatively in a Page-type experiment, even with little improvement in the accuracy he achieved in 1950.