My professor mentioned that a particle with an intrinsic magnetic quadrupole moment would be CP violating in an analogous manner to how a particle with an electric dipole would be evidence for new sources of CP violation. Are then any active or past experiments which are trying to measure the magnetic quadruple moment of particles like the electron for example? I know of at least two ongoing experiments to measure the electric dipole moment of the electron and neutron, but I'm wondering if anyone is studying magnetic quadrupole moments in the same way.
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A spin-half particle cannot have an intrinsic quadrupole moment, because the two-state system does not have enough degrees of freedom. See this answer for a hand-waving explanation, which is related to the Wigner-Eckart theorem.
In a comment, you link to this preprint, which discusses magnetic quadrupole moments in a number of odd-$A$ nuclei with spin $3/2$ or higher.
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