Can the conformal factor of the gravitational field be a physical degree of freedom?

The role of the conformal factor was analysed in two gauge-invariant perturbative formulations. Using the classical and quantum linearized perturbation approach given by Mukhanov et al \cite{Mukhanov:1990me}, the non-physical behaviour of the conformal perturbation as an isolated degree of freedom is shown. In the quantum gravity context, the restriction to conformal perturbations gives us an incompatibility with the dynamical equations.