The Standard Model (SM) predicts an "aspherical" electron due to its possesing a non-zero electron electric dipole moment (EEDM). An experiment by Hinds (2011) placed an upper bound on the asphericity of the electron without absolutely being able to say the EEDM was zero. Most sources I've read somewhat authoritatively say that an electron is a point and thus possesses no volume.
If the SM calls something this small "aspheric", does that logically mean that it predicts the electron possesses what we would call a shape in space and thus a volume? (Even if this shape/volume is a probabality distribution function like atomic orbitals shapes) Does "shape" even have a meaning at these length scales due to QM? (If quantum foam theory is right, wouldn't the electron's shape be continually buffeted and deformed by the foam thus rendering any shape/volume as "indefinite"?) Does the fact that we have nothing smaller than an electron to probe the electron mean that we can't ever know its shape even if it has one?
I know my questions probably indicate I'm helplessly stuck in the classical world, but any insights to help are appreciated.