(Pseudo)issue of the conformal frame revisited

The issue of the equivalence between Jordan and Einstein conformal frames in scalar-tensor gravity is revisited, with the emphasis on implementing running units in the latter. The lack of affine parametrization for timelike worldlines and the cosmological constant problem in the Einstein frame are clarified, and a paradox in the literature about cosmological singularities appearing only in one frame is solved. While, classically, the two conformal frames are physically equivalent, they seem to be inequivalent at the quantum level.