In the previous sections, we have developed the EF formalism to describe the dynamics of a PEN wavefunction with illustrative examples on model systems representing typical situations of nonadiabatic dynamics and spontaneous emission in the strong light-matter coupling regime as achievable in an optical cavity. Building upon previous work of Maitra and coworkers, we have proposed to treat the photonic and the nuclear degrees of freedom on equal footing, i.e., using the marginal amplitude in EF, with their dynamics taking place under the effect of the TDPES and the TDVP that account for the dynamics of the electrons. Applying this formulation of EF to CTMQC, we have treated the photon-nuclear dynamics using coupled trajectories. The comparison between the results of CTMQC simulations with QD simulations combined with the analysis of TDPES and TDVP allowed us to point out the key approximations done in CTMQC that lose validity when extended to treat "light" particles, as the photons in this case. Specifically, the geometric contribution of the TD-PES becomes dominant in determining the shape of the TD-PES, while it is completely neglected in the current formulation of CTMQC based on arguments depending on the small electron-nuclear mass ratio applied to the electron-nuclear formulation of EF . Note that these observations are not limited to the case of PEN systems, but can be extended to other multi-component systems where EF and CTMQC are used, such as systems of protons and heavier nuclei or purely electronic systems. The question now arises as to whether including the geometric part of the TDPES for the photon-nuclear dynamics based on trajectories is sufficient to recover satisfactory performance of CTMQC or the corresponding contribution in the PENCO operator (17) in the electronic evolution equation is also need. Further studies are, however, needed to address this issue. Nonetheless, the numerical applications have shown an overall superior performance of CTMQC in comparison to MTE and TSH when applied to the examples proposed in Section III, even though we are well aware that these are simple model studies and that additional investigations are needed to generalize this observation.