  The issue of the equivalence between Jordan and Einstein conformal frames in
scalar-tensor gravity is revisited, with 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.
