  Seeking a possible explanation for recent data indicating a space-time
variation of the electron-to-proton mass ratio within the Milky Way, we
consider a phenomenological model where the effective fermion masses depend on
the local value of the Weyl tensor. We contrast the required values of the
model's free parameters with bounds obtained from modern tests on the violation
of the Weak Equivalence Principle and we find that these quantities are
incompatible. This result indicates that the variation of nucleon and electron
masses through a coupling with the Weyl tensor is not a viable model.
